4.4 Solving exponential and logarithmic equations

Inverse properties and logarithm laws allow an unknown exponent or logarithmic argument to be isolated. Every logarithmic candidate must be checked against the original domain.

4.4.1 Exponential equations

If both sides can be written with the same base, equate exponents. Otherwise, isolate the exponential expression, take logarithms, apply the power law, and solve.

Worked Example: Using a common base

Solve \(3^{2x+1}=1\). Since \(1=3^0\), \(2x+1=0\) and \(x=-1/2\).

Try It. Solve \(2^{3x-2}=8\).
Check Your Work \(8=2^3\), so \(3x-2=3\) and \(x=5/3\).

Worked Example: Taking logarithms

For \(3^{2x+3}=5\),

\[(2x+3)\ln3=\ln5,\]

so \(x=[\ln5/\ln3-3]/2\approx-0.768\).

Try It. Solve \(4^{x-1}=17\).
Check Your Work \(x=1+\ln17/\ln4\approx3.044\).

4.4.2 Logarithmic equations and domain checks

Combine logarithms when useful, convert to exponential form, solve the resulting algebraic equation, and reject any value that makes an original logarithmic argument nonpositive.

Worked Example: Solving a logarithmic equation

Solve \(\log_7x+\log_7(x+1)=\log_7(20)\). Then \(x(x+1)=20\), so \((x+5)(x-4)=0\). The domain requires \(x>0\), leaving \(x=4\).

Try It. Solve \(\log_2(x+1)-\log_2(x-1)=2\).
Check Your Work \((x+1)/(x-1)=4\), so \(x=5/3\), which satisfies \(x>1\).

Worked Example: Solving a decay-time equation

If \(C(t)=3(0.88)^t\) and the target is 1 mg/L, then

\[t=\frac{\ln(1/3)}{\ln(0.88)}\approx8.59\ \text{h}.\]

Try It. When does \(10(0.75)^t=2\)?
Check Your Work \(t=\ln(0.2)/\ln(0.75)\approx5.59\) periods.

Applied Problem: Meeting a disinfection residual target. Chlorine residual follows \(C(t)=2.8e^{-0.16t}\) mg/L. Determine when it first falls to 0.50 mg/L and interpret the result.

Check Your Work \(0.50=2.8e^{-0.16t}\) gives \(t=-\ln(0.50/2.8)/0.16\approx10.77\) h. The residual reaches 0.50 mg/L about 10.8 h after the initial measurement.

4.4.3 Practice Problems

  1. Solve \(5^{x+1}=25\).
    Check Your Work \(x=1\).
  2. Solve \(2^x=7\).
    Check Your Work \(x=\ln7/\ln2\approx2.807\).
  3. Solve \(\ln x=3\).
    Check Your Work \(x=e^3\).
  4. Solve \(\log(x)+\log(x-9)=1\).
    Check Your Work \(x(x-9)=10\) gives \(x=10\) or \(-1\); only \(x=10\) is valid.
  5. Solve \(\ln(2x-1)=\ln7\).
    Check Your Work \(2x-1=7\), so \(x=4\).
  6. A contaminant follows \(M(t)=40(0.82)^t\). When is \(M=10\)?
    Check Your Work \(t=\ln(0.25)/\ln(0.82)\approx6.99\) periods.
  7. A bacterial count follows \(N(t)=500e^{0.12t}\). When does it reach 1200?
    Check Your Work \(t=\ln(1200/500)/0.12\approx7.30\) h.
  8. Solve \(\log(C)+\log(C-3)=\log28\) for a positive concentration.
    Check Your Work \(C(C-3)=28\), so \((C-7)(C+4)=0\). The valid solution is \(C=7\).