Math 150 – Fall 2015
Section 6A
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Section 6A – Exponential and Logarithmic
Equations
Solving Exponential Equations: To solve an exponentional equation, use the following steps:
1. Isolate the expression containing the exponent on one side of the equation.
2. Take the logarithm of both sides to “bring down the exponent.”
3. Solve for the variable.
4. Be careful: we cannot take a logarithm of 0 or a negative number!
Example 1. Solve the following equations.
(a) 10 − 11x = 7
(b) 5e3x
(c)
1 x
2
8−e2x
5e2x +2
− 3e3x = 0
=3
Math 150 – Fall 2015
Section 6A
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(d) 3e2x − 14ex + 11 = 0
(e) x2 2x − x2 − 9(2x ) + 9 = 0.
Solving Logarithmic Equations: To solve a logarithmic equation, perform the following steps:
1. Isolate the expression containing the logarithm on one side of the equation.
2. Exponentiate both sides using the same base as the log to remove the log function.
3. Solve for the variable.
4. Check: You must check that the solutions are in the domains of your original
logarithms!
Note. When solving an exponential equation we do not have to use the same base since
ln ax = x ln a for any base a. However, when solving a logarithmic equation, we must
use the same base since alogb x 6= x when a and b are not equal. The fact aloga x = x is
true because ax and loga x are inverses of each other.
Example 2. Solve the following equations for x.
(a) log x = 7
Math 150 – Fall 2015
Section 6A
(b) 7 − ln(2x − 3) = 8
(c) 2 + log3 (x − 1) = log3 (x − 4)
(d) log5 7 + log5 x = log5 4 + log5 (x + 3)
(e) 2 ln x = ln 2 + ln(x + 12)
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