A.11(A)·Algebra II·Exponential & Logarithmic FunctionsReadiness

Exponential and Logarithmic Relationships

Connect numerical patterns to exponential and logarithmic functions.

Students define a logarithm as an exponent, then use swapped table values and reflected graphs to connect $y=b^x$ with its inverse, $y=\log_b x$ .

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

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Learn the concept

A logarithm is an exponent

For $b>0$ and $b\ne 1$ , the statement $\log_b x=y$ means exactly the same thing as $b^y=x$ . The common logarithm uses base 10, so $\log 100=2$ because $10^2=100$ . A logarithm answers the question “what exponent is needed?”

Inverses swap ordered pairs

If $(2,100)$ is on $y=10^x$ , then $(100,2)$ is on $y=\log x$ . The graphs are reflections across $y=x$ . The exponential function has domain all real numbers and range $y>0$ ; the logarithm reverses them, with domain $x>0$ and range all real numbers.

Test a proposed rule

When a table claims to show an inverse, check every pair, not just one. A linear rule such as $10x$ may match a single pair, but only the exponential rule $10^x$ reproduces the repeated multiplication by 10.

Worked example

A table for g shows inputs 1, 2, 3 and outputs 3, 9, 27. Write the inverse of g.

Each output is 3 raised to the input, so $g(x)=3^x$ .
Swap every pair: (3, 1), (9, 2), and (27, 3).
Each swapped pair asks for an exponent on 3; for example, $\log_3 9=2$ because $3^2=9$ .

$g^{-1}(x)=\log_3 x$

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Inverses of Exponential Functions

UT IPSI · Kid2Kid / Texas Education Agency

Connect exponential functions to logarithms by finding and interpreting their inverses.

Watch on YouTube

Texas public school video

Algebra 2 Functions and Their Inverses

UT IPSI · Kid2Kid / Texas Education Agency

Swap inputs and outputs in tables and graphs to identify a function's inverse.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Write $10^{-2}=0.01$ in logarithmic form.
Show answer and reasoning
$\log 0.01=-2$
The exponent, −2, becomes the output of the base-10 logarithm, and the result of the power, 0.01, becomes its input.
2. The point $(1000,3)$ is on $f(x)=\log x$ . What point is on its inverse?
Show answer and reasoning
$(3,1000)$
Inverse functions swap inputs and outputs. The inverse is $y=10^x$ , and $10^3=1000$ .

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 1 · 1 point

Some values for the function $f(x) = \log x$ are given in Table 1. Which function can generate all the values in Table 2?

Table 1 and Table 2
Table	x	Function values
Table 1: $f(x) = \log x$	10, 100, 1,000, 10,000	1, 2, 3, 4
Table 2: $g(x)$	1, 2, 3, 4	10, 100, 1,000, 10,000

Each comma-separated pair lists the x-values and corresponding function values in their original order.

$g(x) = x^{10}$
$g(x) = 10^x$
$g(x) = \frac{10}{x}$
$g(x) = 10x$

Show STAAR answer and explanation

$g(x) = 10^x$

Evaluate $g(x)=10^x$ at 1, 2, 3, and 4 to get 10, 100, 1,000, and 10,000. Every table pair matches.

$g(x) = x^{10}$ : This gives $1^{10}=1$ and $2^{10}=1{,}024$ , not 10 and 100.

$g(x) = \frac{10}{x}$ : This decreases as x increases and gives $g(2)=5$ , not 100.

$g(x) = 10x$ : This gives 10, 20, 30, and 40; the table values multiply by 10 rather than add 10.

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Archived released STAAR Algebra II EOC · 2013 · Item 21 · 1 point

What is the inverse of $h(x) = \frac{1}{2}\log_3(x)$ ?

$h^{-1}(x) = 3^{(2x)}$
$h^{-1}(x) = 2(3)^x$
$h^{-1}(x) = 2(\log_x 3)$
$h^{-1}(x) = \log_3(2x)$

Show STAAR answer and explanation

$h^{-1}(x) = 3^{(2x)}$

Swap x and y: $x=\frac12\log_3 y$ . Multiply by 2: $2x=\log_3 y$ . Rewrite in exponential form: $y=3^{2x}$ .

$h^{-1}(x) = 2(3)^x$ : This multiplies the power by 2 instead of putting $2x$ in the exponent. Since $h(9)=1$ , the inverse must give 9 at $x=1$ , but $2(3)^1=6$ .

$h^{-1}(x) = 2(\log_x 3)$ : This only changes the base of the logarithm. The inverse of a logarithmic function is exponential.

$h^{-1}(x) = \log_3(2x)$ : This is still a logarithm, so it cannot undo $\log_3$ .

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Translate between exponential form $b^y=x$ and logarithmic form $\log_b x=y$ .
Recognize inverse functions in tables by their swapped input-output pairs.
Compare the domains, ranges, and asymptotes of $y=10^x$ and $y=\log x$ .

Instructional sequence

1
Engage
5 min
List 10, 100, 1,000, and 10,000. Ask students what exponent on 10 produces each number.
2
Direct instruction
10 min
Define $\log_b x=y$ as “the exponent on b that produces x.” Introduce the common logarithm, $\log x=\log_{10}x$ .
3
Guided practice
12 min
Complete a table for $y=10^x$ , then swap its columns to build the table for $y=\log x$ . Verify each swapped pair with the definition.
4
Graph connection
15 min
Plot both functions with $y=x$ . Students mark matching reflected points and compare $y=0$ and $x=0$ as asymptotes.
5
Closure
8 min
Students test one proposed inverse rule against every table pair, then complete the exit ticket.

Misconceptions to address

““

\log x

means log multiplied by x.””

Teacher move: Treat log as a function name. Its output is an exponent: $\log 1000=3$ because $10^3=1000$ .

““The inverse of

\log x

x^{10}

\frac{10}{x}

.””

Teacher move: An inverse undoes the original function by swapping each pair. Test the rule: only $10^x$ turns 1, 2, 3 back into 10, 100, 1,000.

Materials

•Paired tables for $y=10^x$ and $y=\log x$
•Coordinate grids with the line $y=x$
•Scientific or graphing calculator

Teaching tips

Have students say every logarithm as a question: “10 to what power gives this number?”
Require a check of at least three table pairs before accepting any proposed function rule.
Color-code inputs and outputs so the column swap between $10^x$ and $\log x$ is visible.

Exit ticket · 3–5 min

Rewrite $\log_2 32=5$ in exponential form. Which point on $y=2^x$ matches the point $(32,5)$ on $y=\log_2 x$ ?

Show expected answer

$2^5=32$ . Swapping the pair gives $(5,32)$ on $y=2^x$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

A logarithm is an exponent

Inverses swap ordered pairs

Test a proposed rule

Worked example

Watch the lesson

Algebra 2 Inverses of Exponential Functions

Algebra 2 Functions and Their Inverses

Try it yourself

1. Write 10−2=0.0110^{-2}=0.0110−2=0.01 in logarithmic form.

2. The point (1000,3)(1000,3)(1000,3) is on f(x)=log⁡xf(x)=\log xf(x)=logx. What point is on its inverse?

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 1 · 1 point

Archived released STAAR Algebra II EOC · 2013 · Item 21 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Graph connection

Closure

Misconceptions to address

1. Write $10^{-2}=0.01$ in logarithmic form.

2. The point $(1000,3)$ is on $f(x)=\log x$ . What point is on its inverse?