Solving exponential and logarithmic equations
Logarithms bring variables down from exponents, and exponentials undo logarithms. Together they solve equations about growth, decay, and compound interest.
Same base
If both sides can be written with the same base, set the exponents equal: \(2^{x + 1} = 32 = 2^5\), so \(x + 1 = 5\) and x = 4. Similarly, \(9^x = 27\) becomes \(3^{2x} = 3^3\), so \(x = \tfrac{3}{2}\).
Taking logarithms
Otherwise, take the log of both sides and use the power rule. For \(5^x = 40\): \(x\ln 5 = \ln 40\), so \(x = \frac{\ln 40}{\ln 5} \approx \frac{3.689}{1.609} \approx 2.29\). Check: \(5^{2.29} \approx 40\).
Isolate the exponential first
For \(3e^{2x} - 5 = 16\), isolate: \(e^{2x} = 7\). Take natural logs: \(2x = \ln 7\), so \(x = \frac{\ln 7}{2} \approx 0.973\). Taking logs before isolating leads to the log of a sum, which has no rule.
Logarithmic equations
To solve \(\log_2(x + 3) = 5\), rewrite in exponential form: \(x + 3 = 2^5 = 32\), so x = 29. If several logs appear, combine them first: \(\log x + \log(x - 3) = 1\) becomes \(\log[x(x - 3)] = 1\), so \(x^2 - 3x = 10\) and \((x - 5)(x + 2) = 0\).
Checking the domain
Logs of zero or negative numbers are undefined, so check every answer. In the last example, x = -2 would require \(\log(-2)\), so it is extraneous, and the only solution is x = 5. Check: \(\log 5 + \log 2 = \log 10 = 1\).
Time to reach a target
How long does $2,000 take to grow to $5,000 at 4% compounded annually? \(2000(1.04)^t = 5000\), so \(1.04^t = 2.5\) and \(t = \frac{\ln 2.5}{\ln 1.04} \approx \frac{0.916}{0.0392} \approx 23.4\) years.
Radioactive dating
If a bone has 30% of its original carbon-14, then \(\left(\tfrac{1}{2}\right)^{t/5730} = 0.30\). Taking logs: \(\frac{t}{5730} = \frac{\ln 0.30}{\ln 0.5} \approx 1.737\), so \(t \approx 9{,}950\) years. Willard Libby developed radiocarbon dating in the late 1940s and won the 1960 Nobel Prize in Chemistry.
Worked example
A town of 12,000 grows continuously at 2.5% a year: \(P = 12{,}000e^{0.025t}\). When will it reach 20,000? \(e^{0.025t} = \tfrac{5}{3}\), so \(0.025t = \ln\tfrac{5}{3} \approx 0.511\), giving \(t \approx 20.4\) years.
Key takeaways
If both sides share a base, set the exponents equal. Otherwise isolate the exponential, take logarithms of both sides, and use the power rule to bring the variable down. Solve logarithmic equations by combining logs and rewriting in exponential form. Always reject answers that would take the log of zero or a negative number. These methods find how long investments take to reach a target, how old carbon-dated samples are (Libby, Nobel 1960), and when growing populations reach a size.
Flashcard
Equal bases means equal exponents
Check your understanding
Solve \(\log_2(x + 3) = 5\).