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  2. AP Pre Calculus
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How does multiplying the input of a log function by a constant affect its graph?

Results in a vertical translation (shift up/down) of the graph.

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How does multiplying the input of a log function by a constant affect its graph?

Results in a vertical translation (shift up/down) of the graph.

How does raising the input of a log function to a power affect its graph?

Results in a vertical dilation (stretch or compression) of the graph.

How does changing the base of a logarithmic function affect its graph?

It affects the 'height' of the graph, as all log functions are vertical dilations of each other.

What is a logarithm?

The inverse function to exponentiation. If by=xb^y = xby=x, then log⁡b(x)=y\log_b(x) = ylogb​(x)=y.

What is the natural logarithm?

The logarithm with base e, denoted as ln⁡(x)\ln(x)ln(x).

What is the base of the natural logarithm?

The mathematical constant e, approximately equal to 2.71828.

How do you expand a logarithmic expression using the product property?

Identify products within the logarithm, then split the log into a sum of logs: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)logb​(xy)=logb​(x)+logb​(y).

How do you simplify a logarithmic expression using the power property?

Identify exponents within the logarithm, then bring the exponent down as a multiplier: log⁡b(xn)=nlog⁡b(x)\log_b(x^n) = n\log_b(x)logb​(xn)=nlogb​(x).

How do you change the base of a logarithm to base 10?

Use the change of base formula: log⁡b(x)=log⁡10(x)log⁡10(b)\log_b(x) = \frac{\log_{10}(x)}{\log_{10}(b)}logb​(x)=log10​(b)log10​(x)​.

How do you solve logarithmic equations?

Apply properties to isolate the variable. Rewrite in exponential form if necessary.