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Intermediate Algebra / LEVEL 4 · DIFFICULTY 4/5

Functions and Identities

Simplify logarithmic, absolute-value and symmetric relationships.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST IA 4.1Logarithmic StructureRead the lesson
MINI QUEST IA 4.2Absolute Value as DistanceRead the lesson
MINI QUEST IA 4.3Symmetric ExpressionsRead the lesson
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STAGE IA 4.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Logarithmic Structure

Useful preparation: Bounding Positive Expressions

Goal: Understand and apply logarithmic structure.

Before you begin: Bounding Positive Expressions

PARENT EQUATION

Logarithmic family: $\displaystyle y = \operatorname{log}_{b}\, \left(x\right)\, \left(b > 0, b \ne 1\right)$

The unshifted graph passes through (1, 0). Its domain is $\displaystyle x > 0$ and its vertical asymptote is $\displaystyle x = 0$ .

How does the question’s equation change this graph?

$\displaystyle y = a \cdot \operatorname{log}_{b}\, \left(x - h\right) + k$ shifts the asymptote to $\displaystyle x = h$ and requires $\displaystyle x > h$ . The exponential and logarithm of the same base are inverses.

Check: Check that every logarithm’s input is positive before accepting an answer.

First identify this family. Then read the coefficients and shifts in your actual equation; this reference does not replace the question’s given values.

Understand the idea

A logarithm names an exponent. Rewriting positive arguments as powers of the same base makes the exponent relationships visible and avoids approximate computation.

\displaystyle \operatorname{log}_{b}\, \left(b^{k}\right) = k

\displaystyle b > 0

and

\displaystyle b \ne 1

Choose and carry out a method

Identify the power of the base in each argument. Replace each logarithm with that exponent, then apply the operation requested outside the logs.

Check the reasoning

Logarithm arguments must be positive and the base must be positive and unequal to one. Rules for products do not become rules for sums.

WORKED EXAMPLE 1

Evaluate $\displaystyle \operatorname{log}_{3}\, \left(9\right) + \operatorname{log}_{3}\, \left(3\right)$ .

A logarithm is the exponent needed to obtain its argument.
$\displaystyle 9 = 3^{2}$ and $\displaystyle 3 = 3^{1}$ .
The scores add to $\displaystyle 2 + 1 = 3$ . Both arguments and the base satisfy the logarithm restrictions.

WORKED EXAMPLE 2

Evaluate $\displaystyle \operatorname{log}_{4}\, \left(16\right) + \operatorname{log}_{4}\, \left(4\right)$ .

A logarithm is the exponent needed to obtain its argument.
$\displaystyle 16 = 4^{2}$ and $\displaystyle 4 = 4^{1}$ .
The scores add to $\displaystyle 2 + 1 = 3$ . Both arguments and the base satisfy the logarithm restrictions.

Common pitfalls

Possible mix-up: $\displaystyle \operatorname{log}\, \left(a + b\right) = \operatorname{log}\, a + \operatorname{log}\, b$ .

The addition of logs corresponds to multiplying arguments.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Explain a logarithm by asking a question about an exponent.

Preview the eight practice prompts

Problem 1
Evaluate $\displaystyle \operatorname{log}_{3}\, \left(27\right) + \operatorname{log}_{3}\, \left(3\right)$ .
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Problem 2
Evaluate $\displaystyle \operatorname{log}_{4}\, \left(64\right) + \operatorname{log}_{4}\, \left(4\right)$ .
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Problem 3
Evaluate $\displaystyle \operatorname{log}_{2}\, \left(16\right) + \operatorname{log}_{2}\, \left(2\right)$ .
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Problem 4
Evaluate $\displaystyle \operatorname{log}_{3}\, \left(81\right) + \operatorname{log}_{3}\, \left(3\right)$ .
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Problem 5
Evaluate $\displaystyle \operatorname{log}_{4}\, \left(256\right) + \operatorname{log}_{4}\, \left(4\right)$ .
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Problem 6
Evaluate $\displaystyle \operatorname{log}_{2}\, \left(32\right) + \operatorname{log}_{2}\, \left(2\right)$ .
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Problem 7New context
A scale assigns score $\displaystyle \operatorname{log}_{3}\, \left(v\right)$ to positive value v. Two readings are 243 and 3; what is the sum of their scores?
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Problem 8New context
A scale assigns score $\displaystyle \operatorname{log}_{4}\, \left(v\right)$ to positive value v. Two readings are 1024 and 4; what is the sum of their scores?
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Open stage IA 4.1 in the student workspace →

STAGE IA 4.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Absolute Value as Distance

Useful preparation: Logarithmic Structure

Goal: Understand and apply absolute value as distance.

Before you begin: Logarithmic Structure

Understand the idea

The equation $\displaystyle \left|x - h\right| = r$ places x at distance r from h on a number line. For positive r, there are two symmetric positions, one on each side.

\displaystyle \left|x - h\right| = r \Rightarrow x = h \pm r

for

\displaystyle r \ge 0

Choose and carry out a method

Write $\displaystyle x - h = r$ or $\displaystyle x - h = -r$ . Solve both linear equations and combine the values in the way the question requests.

Check the reasoning

A negative radius would give no real solutions. With radius zero, the two positions coincide and must not be counted twice.

WORKED EXAMPLE 1

The equation $\displaystyle \left|x - 3\right| = 6$ has two real solutions. Find their sum.

Absolute value measures distance, so there is one solution on each side of the center.
$\displaystyle x = 3 - 6$ or $\displaystyle x = 3 + 6$ .
The sum is $\displaystyle 2 \cdot 3 = 6$ ; the distance terms cancel.

WORKED EXAMPLE 2

The equation $\displaystyle \left|x - 4\right| = 7$ has two real solutions. Find their sum.

Absolute value measures distance, so there is one solution on each side of the center.
$\displaystyle x = 4 - 7$ or $\displaystyle x = 4 + 7$ .
The sum is $\displaystyle 2 \cdot 4 = 8$ ; the distance terms cancel.

Common pitfalls

Possible mix-up: Only $\displaystyle x = h + r$ is possible.

A positive distance allows positions on both sides of h.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why is the sum of the two solutions independent of the radius?

Preview the eight practice prompts

Problem 1
The equation $\displaystyle \left|x - 6\right| = 9$ has two real solutions. Find their sum.
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Problem 2
The equation $\displaystyle \left|x - 7\right| = 10$ has two real solutions. Find their sum.
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Problem 3
The equation $\displaystyle \left|x - 8\right| = 11$ has two real solutions. Find their sum.
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Problem 4
The equation $\displaystyle \left|x - 9\right| = 12$ has two real solutions. Find their sum.
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Problem 5
The equation $\displaystyle \left|x - 10\right| = 13$ has two real solutions. Find their sum.
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Problem 6
The equation $\displaystyle \left|x - 11\right| = 14$ has two real solutions. Find their sum.
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Problem 7New context
Two markers on a number line are each 15 units from position 12. What is the sum of their coordinates?
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Problem 8New context
Two markers on a number line are each 16 units from position 13. What is the sum of their coordinates?
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Open stage IA 4.2 in the student workspace →

STAGE IA 4.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Symmetric Expressions

Useful preparation: Absolute Value as Distance

Goal: Understand and apply symmetric expressions.

Before you begin: Absolute Value as Distance

PARENT EQUATION

Quadratic parent: $\displaystyle y = x^{2}$

A U-shaped curve with vertex (0, 0), symmetric about the y-axis.

How does the question’s equation change this graph?

Write $\displaystyle y = a\, \left(x - h\right)^{2} + k$ . Move the vertex to (h, k), stretch vertically by $\displaystyle \left|a\right|$ , and reflect across the x-axis if $\displaystyle a < 0$ . In $\displaystyle ax^{2} + bx + c$ , $\displaystyle h = \frac{-b}{2\, a}$ and $\displaystyle k = c - \frac{b^{2}}{4\, a}$ .

Check: The sign inside the brackets is opposite the horizontal shift: $\displaystyle \left(x - 3\right)^{2}$ moves right 3. This form requires $\displaystyle a \ne 0$ .

First identify this family. Then read the coefficients and shifts in your actual equation; this reference does not replace the question’s given values.

AN EQUATION IS A RULE FOR MAKING POINTS

Change the rule. Watch the graph.

Pick an x value. The rule tells you its y value. A point (x, y) shows both numbers together. Drag across the graph or use the point slider. This is an adjustable learning example; use the coefficients in your own practice question when solving it.

Parent equation: $\displaystyle y = x^{2}$

Start with $\displaystyle y = x^{2}$ . Complete the square: $\displaystyle y = 1\, \left(x - \left(1\right)\right)^{2} + \left(-4\right)$ . The new vertex is (1, -4). The first coordinate gives the horizontal shift and the second gives the vertical shift. Multiply vertical distances from the vertex by 1.

Transform the parent graph

Teal moving graph: $\displaystyle y = 1\, x^{2} + \left(0\right)\, x + \left(0\right)$

Blue: $\displaystyle y = 1\, x^{2} - 2\, x - 3$

Moving point: (-4, 16)

Teal shows the current transformation. Blue is the target. At $\displaystyle 100\%$ they coincide; intermediate graphs are learning frames, not additional solutions.

Dashed gray: parent equation. Blue: current equation. View: −10 to 10 on both axes. The graph continues beyond this window. Purple marks your selected point.

Curve coefficient a: 1

Coefficient b: -2

Vertical intercept c: -3

Choose the point’s x: 1

Point (1, -4)

\displaystyle 1 \times \left(1\right)^{2} + \left(-2\right) \times \left(1\right) + \left(-3\right) = -4

Why does it bend?

The $\displaystyle x^{2}$ term makes the change in y vary with x. This parabola opens upward because a is positive.

Three points made by the blue rule
x	Substitute into the rule	y
-1	$\displaystyle 1 \times \left(-1\right)^{2} + \left(-2\right) \times \left(-1\right) + \left(-3\right)$	0
0	$\displaystyle 1 \times \left(0\right)^{2} + \left(-2\right) \times \left(0\right) + \left(-3\right)$	-3
1	$\displaystyle 1 \times \left(1\right)^{2} + \left(-2\right) \times \left(1\right) + \left(-3\right)$	-4

Understand the idea

Some expressions depend only on the sum and product of two numbers, so finding the individual numbers creates unnecessary work. Squaring the sum exposes the needed cross term.

\displaystyle r^{2} + s^{2} = \left(r + s\right)^{2} - 2\, rs

Choose and carry out a method

Expand $\displaystyle \left(r + s\right)^{2} = r^{2} + 2\, rs + s^{2}$ and rearrange to obtain $\displaystyle r^{2} + s^{2}$ . Substitute the known sum and product directly.

Check the reasoning

This identity holds for real or complex values. Keep the coefficient two on the product term.

WORKED EXAMPLE 1

Numbers r and s satisfy $\displaystyle r + s = 8$ and $\displaystyle rs = 3$ . Find $\displaystyle r^{2} + s^{2}$ .

Expand the square of the sum and subtract the cross-term.
$\displaystyle r^{2} + s^{2} = \left(r + s\right)^{2} - 2\, rs = 8^{2} - 2 \cdot 3$ .
The requested value is 58; individual roots are unnecessary.

WORKED EXAMPLE 2

Numbers r and s satisfy $\displaystyle r + s = 9$ and $\displaystyle rs = 4$ . Find $\displaystyle r^{2} + s^{2}$ .

Expand the square of the sum and subtract the cross-term.
$\displaystyle r^{2} + s^{2} = \left(r + s\right)^{2} - 2\, rs = 9^{2} - 2 \cdot 4$ .
The requested value is 73; individual roots are unnecessary.

Common pitfalls

Possible mix-up: The square of a sum equals the sum of squares.

The expansion includes the cross term 2rs.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Find an expression for $\displaystyle \left(r - s\right)^{2}$ using the same two pieces of information.

Preview the eight practice prompts

Problem 1
Numbers r and s satisfy $\displaystyle r + s = 11$ and $\displaystyle rs = 6$ . Find $\displaystyle r^{2} + s^{2}$ .
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Problem 2
Numbers r and s satisfy $\displaystyle r + s = 12$ and $\displaystyle rs = 7$ . Find $\displaystyle r^{2} + s^{2}$ .
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Problem 3
Numbers r and s satisfy $\displaystyle r + s = 13$ and $\displaystyle rs = 8$ . Find $\displaystyle r^{2} + s^{2}$ .
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Problem 4
Numbers r and s satisfy $\displaystyle r + s = 14$ and $\displaystyle rs = 9$ . Find $\displaystyle r^{2} + s^{2}$ .
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Problem 5
Numbers r and s satisfy $\displaystyle r + s = 15$ and $\displaystyle rs = 10$ . Find $\displaystyle r^{2} + s^{2}$ .
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Problem 6
Numbers r and s satisfy $\displaystyle r + s = 16$ and $\displaystyle rs = 11$ . Find $\displaystyle r^{2} + s^{2}$ .
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Problem 7New context
Two positive lengths have sum 17 and product 12. A model needs the sum of their squares. Find that value without solving for the individual lengths.
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Problem 8New context
Two positive lengths have sum 18 and product 13. A model needs the sum of their squares. Find that value without solving for the individual lengths.
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Open stage IA 4.3 in the student workspace →

Browse another level →

Functions and Identities

Logarithmic Structure

Understand the idea

Choose and carry out a method

Check the reasoning

Evaluate log⁡3 (9)+log⁡3 (3)\displaystyle \operatorname{log}_{3}\, \left(9\right) + \operatorname{log}_{3}\, \left(3\right)log3​(9)+log3​(3).

Evaluate log⁡4 (16)+log⁡4 (4)\displaystyle \operatorname{log}_{4}\, \left(16\right) + \operatorname{log}_{4}\, \left(4\right)log4​(16)+log4​(4).

Common pitfalls

Explain it to yourself

Absolute Value as Distance

Understand the idea

Choose and carry out a method

Check the reasoning

The equation ∣x−3∣=6\displaystyle \left|x - 3\right| = 6∣x−3∣=6 has two real solutions. Find their sum.

The equation ∣x−4∣=7\displaystyle \left|x - 4\right| = 7∣x−4∣=7 has two real solutions. Find their sum.

Common pitfalls

Explain it to yourself

Symmetric Expressions

Change the rule. Watch the graph.

Parent equation: y=x2\displaystyle y = x^{2}y=x2

Understand the idea

Choose and carry out a method

Check the reasoning

Numbers r and s satisfy r+s=8\displaystyle r + s = 8r+s=8 and rs=3\displaystyle rs = 3rs=3. Find r2+s2\displaystyle r^{2} + s^{2}r2+s2.

Numbers r and s satisfy r+s=9\displaystyle r + s = 9r+s=9 and rs=4\displaystyle rs = 4rs=4. Find r2+s2\displaystyle r^{2} + s^{2}r2+s2.

Common pitfalls

Explain it to yourself

Evaluate $\displaystyle \operatorname{log}_{3}\, \left(9\right) + \operatorname{log}_{3}\, \left(3\right)$ .

Evaluate $\displaystyle \operatorname{log}_{4}\, \left(16\right) + \operatorname{log}_{4}\, \left(4\right)$ .

The equation $\displaystyle \left|x - 3\right| = 6$ has two real solutions. Find their sum.

The equation $\displaystyle \left|x - 4\right| = 7$ has two real solutions. Find their sum.

Parent equation: $\displaystyle y = x^{2}$

Numbers r and s satisfy $\displaystyle r + s = 8$ and $\displaystyle rs = 3$ . Find $\displaystyle r^{2} + s^{2}$ .

Numbers r and s satisfy $\displaystyle r + s = 9$ and $\displaystyle rs = 4$ . Find $\displaystyle r^{2} + s^{2}$ .