Exponents and Logarithms
How are equations with exponents and logarithms solved?
The way to solve an exponential equation is relatively easy: Take the log of both sides of the equation, then solve for the variable. For example, to solve for x in the equation ex = 60.
- First, take the natural log (ln) of both sides:
ln(ex) = ln(60)
- Simplify using the logarithmic rule #3 (see above) for the left side:
x ln(e) = ln(60)
- Then simplify again, since ln(e) = 1 to:
x= ln(60) = 4.094344562
- And finally, check your answer (using log tables or your calculator) in the original equation ex = 60:
e 4.094344562 = 60 is definitely true.
The way to solve a logarithmic equation is equally easy: Just rewrite the equation in exponential form and solve for the variable. For example, to solve for x in the equation ln(x) = 11:
- 1.First, change both sides so they are exponents of the base e:
eln(x) = e11
- When the bases of the exponent and logarithm are the same, the left part of the equation becomes x, thus, it can be written:
x = e11
- To obtain x, determine the solution for e11, or
x is approximately 59,874.14172.
- And finally, check your answer (using tables or your calculator) in the original equation ln(x) = 11:
ln(59,874.14172) = 11 is definitely true.