E9. Solving Exponential and Logarithmic Equations
0.1
At this point much practice has been done writing an exponential equation as a
logarithmic equation, writing a logarithmic equation as an exponential equation,
and utilizing the properties of logarithms. Utilizing these tools, both logarithmic
and exponential equations can be solved.
Example. Solve 3x+1 = 2 for x.
Solution.The difficulty of this problem is that the left side is in base 3 and the
right side of the equation is in base 2. There is no whole number base that works
for both sides. Instead of dealing with the exponential, examine its logarithm.
3x+1 = 2 ⇐⇒ log3 2 = x + 1.
Solve the logarithmic equation for x.
log3 2 =x + 1
log3 2 − 1 =x,
or, x = log3 2 − 1.
The solution of x = log3 2−1 is an exact solution. The decimal approximation is
obtained using a calculator to be x ≈ −0.36907. An exact solution that one may
be more familiar with is the number two-thirds, 23 . It’s decimal approximation
is 0.67. The decimals continue forever, so at some value the number is rounded.
A rounded number can be quite close to the exact, but never equal. Other
examples of exact numbers and decimal approximations are e ≈ 2.718 and
π ≈ 3.14.
Example. Solve log(x) = −2.
Solution. The solution is found by writing the logarithmic equation as an
exponential equation.
log(x) = −2 ⇐⇒ 10−2 =x
x =10−2
1
x=
100
or, x =0.01
The ability to move between the exponential and logarithmic equations is the
key to solving these equations.
1
Practice Problems.
Write as an equivalent logarithmic or exponential equation to solve for the
unknown. (Write both the exact answer and the approximate answer to four
decimal places.)
1. 4x = 9
2.
3. ln(x) = −2
4. 3x−1 = 2
5. log x = 2.1
6. log4 (x + 1) =
7. 22x+1 = 5
8. 102x = 0.18
9. log2 (x + 2) = 3
ex = 5
1
2
10. log3 (2x − 1) = 4
11. log(3x − 5) = 2
12. ln(x + 3) = 1.5
13. log4 (3 − x) = 2.7
14.
e2x+2 = 0.04
2
Solutions.
Practice Problems.
Write as an equivalent logarithmic or exponential equation to solve for the
unknown. (Write both the exact answer and the approximate answer to four
decimal places.)
1. 4x = 9 =⇒ x = log4 9 exact solution, and x ≈ 1.5850
Use the change of base formula to approximate log4 9.
2.
ex = 5 =⇒ x = ln 5 exact solution, and x ≈ 1.6094
3. ln(x) = −2 =⇒ x = e−2 exact solution, and x ≈ 0.1353
4. 3x−1 = 2 =⇒ x − 1 = log3 2 =⇒ x = log3 2 + 1
exact solution, and x ≈ 1.6309
5. log x = 2.1 =⇒ x = 102.1 exact solution, and x ≈ 125.8925
6. log4 (x + 1) =
1
2
1
=⇒ 4 2 = x + 1 =⇒ x = 1
7. 22x+1 = 5 =⇒ log2 5 = 2x + 1 =⇒ x =
log2 5−1
2
8. 102x = 0.18 =⇒ log 0.18 = 2x =⇒ x =
1
2
exact solution, and x ≈ 0.6610
log 0.18 exact solution, and x ≈ −0.3724
9. log2 (x + 2) = 3 =⇒ 23 = x + 2 =⇒ x = 6
10. log3 (2x − 1) = 4 =⇒ 34 = 2x − 1 =⇒ x = 41
11. log(3x − 5) = 2 =⇒ 102 = 3x − 5 =⇒ x = 35
12. ln(x + 3) = 1.5 =⇒ e1.5 = x + 3 =⇒ x = e1.5 − 3 exact solution, and x ≈ 1.4817
13. log4 (3 − x) = 2.7 =⇒ 42.7 = 3 − x =⇒ x = 3 − 42.7 exact solution, and x ≈ −39.2243
14.
e2x+2 = 0.04 =⇒ ln 0.04 = 2x + 2 =⇒ x =
3
1
2
ln 0.04 − 1 exact solution, and x ≈ −2.6094