Rewrite this logarithm in exponential form:

y = log(x + 3)

We've got a common log here, so our base is an invisible 10. Let's write that down first:

10

The left 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

Boom, there we go.

Rewrite this exponential equation in logarithmic form:

y = 4(7^x)

First, we should isolate the x by dividing both sides by 4. Then we 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: 

Simplify the following expression:

5^log₅25

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

5^log₅25 = 25

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

One inverse, coming right up. First, we solve for x. Subtract 25 on both sides:

y – 25 = 15x

Divide by 15:

Now switch the variables:

What is the inverse function of y = 10^x – 4?

Add 4 to both sides: 

y + 4 = 10^x

Take the common logarithm of both sides, with 10 as the base:

log(y + 4) = log 10^x

log(y + 4) = x

Switch x and y: 

y = log(x + 4)

Nailed it.

Simplify:

3^log₃4

The base-3 log cancels out the 3, which just leaves us with the exponent as our answer:

3^log₃4 = 4

That's it.

Does this function have an inverse?

f(x) = x² + 3x + 4

If you graph this function, you'll get a parabola. This shape fails the horizontal line test, which means that it's not one-to-one.

Since it's not one-to-one, it can't have an inverse. Point, set, match.