Linking Exponents and Logarithms Examples

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

Show Next Step

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:

Show Next Step

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

Show Next Step

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:

Show Next Step

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)

Show Next Step

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:

Show Next Step

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

Show Next Step

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.

Show Next Step

People who Shmooped this also Shmooped...