Topic 2.10 Notes – Inverses of Exponential Functions

What Logarithmic Functions Are

A logarithm is an exponent. If

$$y=\log_b x$$

that means

$$b^y=x.$$

So $\log_b x$ asks, “What exponent on base $b$ gives me $x$?”

The core inverse pair in this topic is

$$g(x)=b^x \qquad \text{and} \qquad f(x)=\log_b x$$

with $b>0$ and $b\neq 1$.

Why those base rules matter:

• $b>0$ keeps $b^x$ a real exponential function for all real $x$
• $b\neq 1$ matters because $1^x=1$, which is constant, so it cannot have an inverse

You may also see the general logarithmic form

$$f(x)=a\log_b x,$$

but the main inverse relationship here is the basic pair $b^x$ and $\log_b x$, where the exponential has initial value $1$.

Their domain and range swap:

• $y=b^x$ has domain all real numbers, range $(0,\infty)$
• $y=\log_b x$ has domain $(0,\infty)$, range all real numbers

Anchor points you should know cold:

• $(0,1)$ is always on $y=b^x$
• $(1,0)$ is always on $y=\log_b x$

It helps to see those facts all at once, especially the swapped domain and range, the asymptotes, and the reflection across $y=x$.

Exponential and logarithmic inverse graphs

How to Construct the Inverse

From the equation

Take the exponential and reverse input/output:

1. Write $y=b^x$
2. Switch $x$ and $y$, giving $x=b^y$
3. Rewrite in logarithmic form as $y=\log_b x$

So

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

Equivalent forms are the same relationship written two ways:

$$y=b^x \Longleftrightarrow x=\log_b y$$

Example:

$$7^u=v \Longleftrightarrow \log_7 v=u$$

From points or a table

If $(s,t)$ is on $y=b^x$, then $(t,s)$ is on $y=\log_b x$. You swap every ordered pair completely.

For $y=3^x$:

• Exponential points
  $\left(-2,\tfrac19\right),\left(-1,\tfrac13\right),(0,1),(1,3),(2,9)$
• Logarithmic points
  $\left(\tfrac19,-2\right),\left(\tfrac13,-1\right),(1,0),(3,1),(9,2)$

From the graph

Inverse graphs are reflections across the line $y=x$. Every point $(s,t)$ becomes $(t,s)$.

What the Inverse Relationship Looks Like

The strongest way to show two functions are inverses is composition:

$$b^{\log_b x}=x \qquad \text{for } x>0$$

$$\log_b(b^x)=x \qquad \text{for all real } x$$

These are not random log rules. They say the functions undo each other.

Graph features also reflect:

• exponential asymptote $y=0$ becomes logarithmic asymptote $x=0$
• if $b>1$, both functions are increasing
• if $0<b<1$, both are decreasing

Example with a base between 0 and 1:

$$\left(\tfrac12\right)^{-3}=8 \Longleftrightarrow \log_{1/2} 8=-3$$

That one gets missed a lot because students expect logs to increase. They do only when $b>1$.

How Exponential and Logarithmic Change Compare

For the exponential $g(x)=b^x$, adding to the input multiplies the output:

$$g(x+k)=b^k g(x)$$

especially

$$g(x+1)=bg(x).$$

For the logarithm $f(x)=\log_b x$, multiplying the input adds to the output:

$$f(b^k x)=f(x)+k$$

especially

$$f(bx)=f(x)+1.$$

That’s the inverse pattern:

• Exponential growth: additive input changes give multiplicative output changes
• Logarithmic growth: multiplicative input changes give additive output changes

Example with base 4:

• $\log_4 4=1$
• $\log_4 16=2$
• $\log_4 64=3$

Each time the input is multiplied by $4$, the output goes up by $1$.

Common Mistakes and Fast Checks

• Inverse vs reciprocal. If $g(x)=b^x$, then $g^{-1}(x)=\log_b x$, not $\frac1{g(x)}=b^{-x}$.
• Log domain. You can only plug positive inputs into $\log_b x$.
• Wrong reflection line. Inverses reflect across $y=x$.
• Incomplete swapping. $(s,t)$ must become $(t,s)$, not just “move the numbers around.”
• Base between 0 and 1. The functions are still inverses even though both decrease.