# How Logarithmic and Exponential Functions are related?

## How Logarithmic and Exponential Functions are Related - How it Works - Video

**Definition of Log**_{a}

**Definition of Log**

_{a}

Definition of log_{a}:

Remember one definition of logarithms is how many of this number do we multiply to get that number using powers?

Let's take a quick look at 2 * 2 * 2 = 8. We can rewrite 2 * 2 * 2 as 2^{3} because of rules of exponents. Now we have 2^{3} = 8. 2^{3} is our exponential. Now looking back our definition of logarithms. It takes 3 threes multiplied together to get 8 or log_{2}(8) = 3.

**Theorem: Logarithmic Functions are One-to-One**

**Theorem: Logarithmic Functions are One-to-One**

Theorem: Logarithmic Functions are One-to-One:

This theorem states that logarithms are one-to-one, which means they have have an inverse, which is an exponential function. The reason that it is one-to-one is that two different inputs will never have the same output, and if the two outputs are the same, then the inputs must be the same.

**Definition of Logarithm and Natural Logarithm**

**Definition of Logarithm and Natural Logarithm**

Definition of Logarithm and Natural Logarithm:

With logarithms, the standard base is 10. With natural logarithms, the standard base is e.

**Example 1**

**Example 1**

Example 1:

Remember one definition of logarithms is how many of this number do we multiply to get that number using powers?

When converting from either logarithmic functions to exponential functions or vice versa, it is useful to remember that because log_{a}x = y is the same as a^{y} = x. Here we want to change the logarithm to exponential. So the input x will be the result in the exponential. The base will remain the same, and the answer will be the power in the exponential.

a) log_{5} 25 = 2

5^{2} = 25

b) log(y) = 13w

log_{10}(y) = 13w

10^{13w} = y

c) ln(3p) = -2

ln_{e}(3p) = -2

e^{-2} = 3p

d) ln(z) = 3y + 7

ln_{e}(z) = 3y + 7

e^{3y+7} = z

From the definition is log does not have a base, then the base is 10, and the base of ln is always e. Since our current number system is off of base-10, we don't write the 10 because it is understand and math people want to write the least amount. This is one of the reasons that we don't write the base for ln. Anytime we can save time we do.

Example 2:

Remember one definition of logarithms is how many of this number do we multiply to get that number using powers?

When converting from either exponential functions to logarithmic functions or vice versa, it is useful to remember that because a^{y} = x is the same as log_{a}x = y. Here we want to change the exponential to logarithm. So the power will be the result in the logarithm. The base will remain the same, and the answer will be the input in the logarithm.

a) 10^{x} = 10,000

log_{10}(10,000) = x

10^{x} = 10,000= 25

log (10,000) = x

b) 7^{2x} = 2y + 1

log_{7}(2y + 1) = 2x

7^{2x} = 2y + 1

log_{7}(2y + 1) = 2x

c) e^{3k} = -4

ln_{e}(-4) = 3k

e^{3k} = -4

no solution b/c input â‰ negative

d) e^{m*m} = 3c - p

ln_{e}(3c - p) = m^{2}

e^{m}^{^2} = 3c - p

ln(3c - p) = m^{2}

Sometimes, when we do math, we just do the motions. But, we have to careful like with c). Everything looks like it follows the definition, but our input value for natural log or log needs to be greater than 0. Also, if we take a second, a standard exponential function can never be negative.

Once again, from the definition is log does not have a base, then the base is 10, and the base of ln is always e. Since our current number system is off of base-10, we don't write the 10 because it is understand and math people want to write the least amount. This is one of the reasons that we don't write the base for ln. Anytime we can save time we do.