Section 4.4

Remember:

Definition 51

• The common logarithm is \(\log_{10}(x)=\log(x)\).

• The natural logarithm is \(\log_{e}(x)=\ln(x)\).

Theorem 6 (Change Base Formula)

For any positive real number \(x\), \(a\), and \(b\) where \(a\ne 1\) and \(b\ne 1\), the following holds true.

\[\log_a(x)=\frac{\log_b(x)}{\log_a(x)}\]

Consider, \(2^x=7\) if and only if \(\log_2(7)=x\). However, most simple scientific calculators are not able to compute this number. With the change base formula, we have:

\[\log_2(7)=\frac{\ln(7)}{\ln(2)}\]

That is, \(\log_2(7)\approx 2.80735\) and \(\frac{\ln(7)}{\ln(2)} \approx 2.80735\).