Covers the Change of Base formula for evaluating logarithms with any base, and reviews essential identities like log of 1, log of the base, and inverse properties.
For any positive numbers \(M\), \(b\), and \(c\) (where \(b \neq 1\) and \(c \neq 1\)), the formula is: \(\log_b(M) = \frac{\log_c(M)}{\log_c(b)}\). The new base \(c\) can be any valid base, but is usually \(10\) or \(e\) for calculator use.
Use the Change of Base Formula to write \(\log_7(15)\) in terms of common logarithms (base 10).
$$\frac{\log(15)}{\log(7)}$$
Use the Change of Base Formula to write \(\log_3(x)\) in terms of natural logarithms (base e).
$$\frac{\ln(x)}{\ln(3)}$$
Why is the Change of Base Formula useful?
Most standard calculators only have buttons for the common logarithm (\(\log\)) and the natural logarithm (\(\ln\)). The formula allows you to calculate the value of a logarithm with any base using one of those two functions.
What is the value of \(\log_b(1)\) for any valid base \(b\)?
\(0\), because any valid base \(b\) raised to the power of \(0\) equals \(1\) (\(b^0 = 1\)).
What is the value of \(\log_b(b)\) for any valid base \(b\)?
\(1\), because any valid base \(b\) raised to the power of \(1\) equals itself (\(b^1 = b\)).
Simplify the expression \(\log_b(b^x)\).
\(x\). This is an inverse property. The logarithm base \(b\) 'undoes' the exponential base \(b\).
Simplify the expression \(b^{\log_b(x)}\).
\(x\). This is also an inverse property. An exponential base \(b\) 'undoes' a logarithm with base \(b\).
Simplify: \(\log_5(5^3)\).
$$3$$
Simplify: \(10^{\log(7)}\).
$$7$$
Calculate the approximate value of \(\log_2(10)\) using your calculator.
Using the Change of Base Formula: \(\log_2(10) = \frac{\log(10)}{\log(2)} = \frac{1}{\log(2)} \approx 3.32\). Or, you could use \(\frac{\ln(10)}{\ln(2)}\).
