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# 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.

1. First, take the natural log (ln) of both sides:

ln(ex) = ln(60)

2. Simplify using the logarithmic rule #3 (see above) for the left side:

x ln(e) = ln(60)

3. Then simplify again, since ln(e) = 1 to:

x= ln(60) = 4.094344562

4. 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. 1.First, change both sides so they are exponents of the base e:

eln(x) = e11

2. 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

3. To obtain x, determine the solution for e11, or

x is approximately 59,874.14172.

4. 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.

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