You might be wondering how Expo and Log got together in the first place. It couldn't just be Expo's charming good looks, right? In this section, you're going to learn all about how logarithms and exponents are connected—how they're the same and how they're total opposites of one another.

What's a Log, Anyway? Before we get into the nitty-gritty of how logs work, let's take a gander just at how they look. Here's one:

log_{10} x

Weird, right? Why is there a number below the log? What's the log doing to the x ? Sometimes you've got to know what's inside to know how something works. Just like in biology class, except as we cut into exponents and logarithms it won't be nearly as squishy or disgusting. We promise.

Practice:

Example 1
Rewrite this logarithm in exponential form: y = log (x + 3)

The base of this log is 10, so we first write that: 10

The left hand side of the equation is the exponent that goes over the base:

10^{y}

The terms inside the log form the other side of the equation: 10^{y} = x + 3

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Example 2
Rewrite this exponential in logarithmic form: y = 4(7^{x} )

First, we should isolate the x : 0.25 y = 7^{x}

Now take the log of both sides, with the base matching the base of the right hand side:

The log and base on the right hand side cancel:

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Example 3
Simplify the following expression: y = 5^{log525}

A base raised to a logarithmic exponent is the same as the other way around, so they cancel: y = 25

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Example 4
What is the inverse function of y = 15x + 25?

First, we must solve for x. Subtract 25 on both sides: y – 25 = 15x Divide 15:

Now switch the variables:

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Example 5
What is the inverse function of y = 10^{x} – 4?

Add 4 on both sides: y + 4 = 10^{x}

Take the logarithm of both sides: log(y + 4) = x

Switch x and y : y = log(x + 4)

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Example 6
What is the inverse function from the set {1,5 ; 2,8 ; 3,11}?

Using the slope-intercept formula, you can recognize this as the function: y = 3x + 2

Solve for x : y – 2 = 3x

Switch x and y:

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Example 7
Simplify:

y = 9^{log34}

First, change the base so that it has an exponent the same as the log in the exponent:

y = 3^{3log34} A base raised to a logarithmic exponent is the same as the other way around, so they cancel: y = 3^{4} Solve: y = 81

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Example 8
Does this equation have an inverse: y = x ^{2} + 3x + 4?

If you graph this function, you will find that it is a parabola

This shape fails the horizontal line test, this means that it is not one-to-one Because it is not one-to-one, it cannot have an inverse.

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Example 1
What is the inverse function of y = log 4x ?

Answer

Example 2
What is the inverse function of y = ln x ?

Gimme a Hint
Hint

Which irrational number does ln have in its base?

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Example 3
What is the inverse function of y = 15x + 3?

Gimme a Hint
Hint

Solve for the dependent variable first.

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Example 4
What is the inverse function of (5xy + x )^{2} = x ^{4}

Gimme a Hint
Hint

Solve the exponents first.

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Example 5
Is x ^{-1} equivalent to

Gimme a Hint
Hint

What is the notation for an inverse function?

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Yes! Something raised to the -1 power is the reciprocal, not the inverse function.

Example 6
Is y = 4^{x} the inverse of log_{x} 4 = y

Gimme a Hint
Hint

What does the base of the logarithm represent?

Show Answer Answer

No. The inverse of y = 4^{x} is log_{4} x = y. The base of the logarithm is the base of the exponential function.

Example 7
If the inverse of a function y = f (x ) is y = g (x ), what does f (g (x )) and g (f (x )) both equal?

Gimme a Hint
Hint

Try plugging values of x into a function and its inverse.

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Example 8
What is the inverse function of log_{4} (3x + 2)?

Gimme a Hint
Hint

Which number do you have to exponentiate with?

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Example 9
What is the inverse function from the set {1,4; 3,10 ; 6,19}?

Gimme a Hint
Hint

Try finding the function first.

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Example 10
Is the following function one-to-one: x^{2} ?

Gimme a Hint
Hint

Will squaring -2 and 2 (or the positive/negative values of any number) give the same answer?

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Example 11
What is the log form of the exponential function y = 4^{x} – 3?

Gimme a Hint
Hint

It's like finding the inverse, but not quite.

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Example 12
What is the log form of the exponential function y = 10e ^{x} ?

Gimme a Hint
Hint

It's au naturel, right?

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Example 13
What is the inverse function of y = 7^{x + 10} ?

Gimme a Hint
Hint

Don't worry, exponentiation won't hurt the exponent on the 7.

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Example 14
Is the following function one-to-one: x ^{3} + 4?

Example 15
Which domains can this function be one-to-one in? 2x ^{2} – 4

Gimme a Hint
Hint

When will the horizontal line test pass?

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