Logarithmic Functions

Definition: A logarithmic function is a function in the form:

png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Clarge f%28x%29%3Dlog_%7Bb%7Dxwhere b > 0 and b ≠ 1.log

The inverse of exponential functions, logarithms (logs) are another way to write exponents.png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Clarge %5C%5Cy %3Dlog_%7Bb%7Dx %5Cleftrightarrow x %3D b%5E%7By%7D.png

When the base of a log function is 10, it is often written without a base.

When the base of a log function is e, it is known as a natural logarithmic function:

png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Clarge f%28x%29%3Dlog_%7Be%7Dx%3D lnx.png

Looking for properties of logarithms or log rules? See our Properties of Logarithms page.

Example: Solve for a:

png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Clarge log_%7B%5C%3A4%7D%5C%3Aa %3D -2.png


Step 1: Keeping the following rule in mind, this equation can be rewritten to help us solve for a:

png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Clarge y %3Dlog_%7Bb%7Dx %5Cleftrightarrow x %3D b%5E%7By%7D%5C%5C-2 %3D log_%7B%5C%3A4%7D%5C%3Aa%5Cleftrightarrow a %3D 4%5E%7B-2%7D.pngStep 2: Simplify to find a:

png.latex-%5Cdpi%7B150%7D %5Cbg_white %5Clarge %5C%5Ca %3D 4%5E%7B-2%7D %3D %5Cfrac%7B1%7D%7B16%7D%5C%5Clog_%7B4%7D%5Cfrac%7B1%7D%7B16%7D%3D-2.png

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