15 Section 3.4: What a logarithm is
Example 1
Rewrite the following exponentials equations as logarithms.
a. [latex]10^x = 100[/latex]
b. [latex]e^{-3} = 0.0497[/latex]
c. [latex]10^{c} = b[/latex]
d. [latex]3e^{7x} = 12[/latex]
Show Solution
a. [latex]\log(100) = x[/latex]
b. [latex]\ln(0.0497) = -3[/latex]
c. [latex]\log(b) = c[/latex]
d. First we isolate the exponential term by dividing by 3 on both sides, so [latex]e^{7x} = 4[/latex]. Now rewriting in logarithmic form, [latex]\ln(4) = 7x[/latex].
Example 2
Rewrite the following logarithms equations as exponentials
a. [latex]\log(15) = 1.176[/latex]
b. [latex]\ln(29) = 3.367[/latex]
c. [latex]\ln(a) = b[/latex]
Show Solution
a. [latex]10^{1.176} = 15[/latex]
b. [latex]e^{3.367} = 29[/latex]
c. [latex]e^{b} = a[/latex].
Example 3
Evaluate the following without a calculator
a. [latex]\log(10000)[/latex]
b. [latex]\log(10^{-3})[/latex]
c. [latex]\log(\frac{10^{-5}}{10^{-8}})[/latex]
d. [latex]\ln(e^7)[/latex]
e. [latex]\ln(\sqrt{e})[/latex]
f. [latex]10^{\log(5)}[/latex]
g. [latex]\log(\frac{1}{10^{1/6}})[/latex]
h. [latex]2e^{\ln(7)}[/latex]
Show Solution
a. [latex]\log(10000) = 4.[/latex]
b. [latex]\log(10^{-3}) = -3.[/latex]
c. [latex]\log(\frac{10^{-5}}{10^{-8}}) = \log(10^{3}) = 3.[/latex]
d. [latex]\ln(e^7) = 7[/latex]
e. [latex]\ln(\sqrt{e}) = ln(e^{\frac{1}{2}}) = \frac{1}{2}.[/latex]
f. [latex]10^{\log(5)} = 5[/latex]
g. [latex]\log(\frac{1}{10^{1/6}}) = \log(10^{-\frac{1}{6}}) = -\frac{1}{6}.[/latex]
h. [latex]2e^{\ln(7)} = 2 \cdot 7 = 14.[/latex]