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# Exponential and Logarithmic Functions

## Definitions

Exponential Functions

An exponential function has the form:

$$f(x)=b^x$$ or $$y=b^x$$,

where $$b>0$$, $$b \ne 1$$, and $$x$$ is any real number (called exponent).

The value $$b$$ is a fixed number and is often called the base and the function is:

• increasing (or representing growth) when $$b>1$$
• decreasing (or representing decay) when $$0<b<1$$

Logarithmic Functions

The inverse of the exponential function $$y=b^x$$ is the exponential equation $$x=b^y$$, which we obtain by interchanging $$x$$ and $$y$$. We can write this equation in its logarithmic form to express it as a logarithmic function $$y=log_b(x)$$ (read as "log base $$b$$ of $$x$$"), where $$b>0$$, $$b \ne 1$$, and $$x>0$$ is any real number. In this case, the value $$b$$ is called the base, $$x$$ is called the power, and $$y$$ is called the exponent.

So, to summarize, we have the two equivalent forms of a logarithmic function:

 Logarithmic Form $$y=log_b(x)$$ Exponential Form $$x=b^y$$

Although logarithms can have any positive value for its base, there are two bases that are more useful in calculations involving logarithms, especially when using our calculators:

Common Logarithm

The base-$$10$$ logarithm is known as the common logarithm since it is commonly used in many real-life applications such as the pH scale, Richter scale (earthquake intensity) and decibel scale (sound intensity).

It is abbreviated as $$log$$ (i.e. if no base is written, then it is implied to be the common logarithm). For example, $$log_{10}(3)=log(3)$$. To calculate the common logarithm of any number, use the "LOG" button on a scientific calculator.

Natural Logarithm

The mathematical constant $$e \approx 2.171828$$ also appears in various business, economics, and biology applications involving exponential/logarithmic functions, so the base-$$e$$ logarithm also gets its own name, known as the natural logarithm

It is abbreviated as $$ln$$. For example, $$log_e(2)=ln(2)$$. To calculate the natural logarithm of any number, use the "LN" button on a scientific calculator.

## Relationship between Logarithmic Exponential Functions

As we saw above, there are two equivalent forms that represent the same logarithmic function, the logarithmic form and exponential form. So since a logarithm is an exponent (i.e. $$log_b(x)=y \iff b^y=x$$), the properties/rules of logarithms and exponents are similar (To review these rules, see Exponent Rules and Logarithm Rules).

Since the concepts of exponential and logarithmic functions can be confusing at times, a good way to differentiate between the two is to remember what question each function answers:

 For exponential functions, we answer "What will the result be if we raise a number to a certain power?". Example: If the function is $$y=2^x$$, then when $$x=3$$, we have the result of $$y=2^3=8$$.
 For logarithmic functions, we answer "What exponent do we need to raise a number to, in order to get a certain value?". Example: If the function is $$y=log_2(x) \iff 2^y=x$$, then when $$x=8$$, so the exponent we're looking for is $$y=3$$.