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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?".

ExampleIf 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 \(y=3\).

 

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Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.
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