Guided Example #4 - Finding Different Bases

When we were studying exponential functions, we used the property that stated if $b > 0$ $b>0$ and $b \neq 1$ $b\ne1$ and $b^{x} = b^{y}$ $b^x=b^y$ , then $x = y$ $x=y$ . This property applies to problems when the bases on both sides of the equation are the same. We are now going to consider exponential equations where the bases are not the same and use logarithms to solve them.

Example 1

Solve $2^{x} = 6$ $2^x=6$

$\log_{10} 2^{x} = \log_{10} 6$ $\log_{10}2^x=\log_{10}6$

Take the logarithm with base 10 of each side of the equation

$x l o g_{10} 2 = \log_{10} 6$ $xlog_{10}2=\log_{10}6$

Apply Log Rule 3

$x = (\frac{\log_{10} 6}{\log_{10} 2})$ $x=\left(\frac{\log_{10}6}{\log_{10}2}\right)$

Divide both sides of the equation by $\log_{10} 2 x \approx 2.5850$ $\log_{10}2x\approx2.5850$

By calculator

From this example, not that the logarithm of 6 with base 2 can be written as a quotient of the common log of 6 and the common log of 2 or $\log_{2} 6 = (\frac{\log_{10} 6}{\log_{10} 2})$ $\log_26=\left(\frac{\log_{10}6}{\log_{10}2}\right)$ . This leads us to a Change of Base Formula for Logs.

Change of Base Formula: If a and b are positive numbers other than 1, then for all positive real numbers x, $\log_{b} x = (\frac{\log_{a} x}{\log_{a} b})$ $\log_bx=\left(\frac{\log_ax}{\log_ab}\right)$

Example 2

Find $LaTeX: \log_37$

$\log_37$

$LaTeX: \log_37=\left(\frac{\log7}{\log3}\right)\approx1.7712$

$\log_37=\left(\frac{\log7}{\log3}\right)\approx1.7712$