Evaluating ln of a Power of e

Evaluate $\ln(e^5)$ without a calculator.

Rewrite as a question: $\ln(e^5)$ asks: to which power must I raise $e$ to get $e^5$? = $e^{?} = e^5$

Read off the exponent: The power that turns $e$ into $e^5$ is $5$ = $? = 5$

Use the inverse rule: $\ln$ undoes $e^x$: $\ln(e^x) = x$ = $\ln(e^5) = 5$

Answer: $\ln(e^5) = 5$

Switching Between ln Form and Exponential Form

(a) Write $\ln x = 3$ in exponential form and find $x$. (b) Write $e^0 = 1$ in logarithmic form.

Use the definition for (a): $\ln x = y$ means $e^y = x$, so $\ln x = 3$ means $e^3 = x$ = $x = e^3$

Evaluate with a calculator: $e^3 \approx 20.09$ = $x \approx 20.09$

Rewrite (b) in ln form: In $e^0 = 1$ the exponent is $0$ and the result is $1$, so $\ln 1 = 0$ = $\ln 1 = 0$

Answer: (a) $x = e^3 \approx 20.09$. (b) $\ln 1 = 0$.

Simplifying with the Inverse Rules

Simplify $e^{\ln 7}$ and $\ln \sqrt{e}$.

Simplify $e^{\ln 7}$: $e^x$ undoes $\ln$: $e^{\ln x} = x$ for $x > 0$ = $e^{\ln 7} = 7$

Write $\sqrt{e}$ as a power of $e$: A square root is the power $\frac{1}{2}$ = $\sqrt{e} = e^{1/2}$

Apply $\ln(e^x) = x$: $\ln(e^{1/2})$ is the exponent $\frac{1}{2}$ = $\ln \sqrt{e} = \frac{1}{2}$

Answer: $e^{\ln 7} = 7$ and $\ln \sqrt{e} = \frac{1}{2} = 0.5$

Evaluating ln with a Calculator

Find $\ln 10$ to 3 decimal places and check the answer.

Press the LN key (not LOG): Type ln(10) on the calculator = $\ln 10 = 2.302585...$

Round to 3 decimal places: The 4th decimal is $5$, so round up = $\ln 10 \approx 2.303$

Check with $e^x$: $e^{2.303} \approx 10.004$, which is very close to $10$ = Check passed

Make sense of the size: $e^2 \approx 7.39$ and $e^3 \approx 20.09$, and $10$ lies between them = $2 < \ln 10 < 3$

Answer: $\ln 10 \approx 2.303$

Solving an Exponential Equation with ln

Solve $3e^{2x} = 60$. Round to 3 decimal places.

Isolate the power of $e$: Divide both sides by $3$: $e^{2x} = \frac{60}{3}$ = $e^{2x} = 20$

Take $\ln$ of both sides: $\ln(e^{2x}) = \ln 20$, and $\ln(e^{2x}) = 2x$ = $2x = \ln 20$

Evaluate $\ln 20$: Calculator: $\ln 20 \approx 2.9957$ = $2x \approx 2.9957$

Divide by 2: $x = \frac{\ln 20}{2} \approx 1.4979$ = $x \approx 1.498$

Check: $3e^{2 \times 1.498} = 3e^{2.996} \approx 3 \times 20.005 \approx 60$ = Check passed

Answer: $x = \frac{\ln 20}{2} \approx 1.498$

Mistake: Writing $\ln 1 = 1$ or $\ln e = 0$

Why: The two key values are easy to swap. $\ln$ gives an exponent, not the number itself.

Correct: $\ln 1 = 0$ because $e^0 = 1$, and $\ln e = 1$ because $e^1 = e$.

Mistake: Taking $\ln$ of zero or a negative number, like $\ln 0$ or $\ln(-3)$

Why: $e^y$ is always positive, whatever $y$ is. No power of $e$ gives $0$ or a negative number.

Correct: $\ln x$ only exists for $x > 0$. The calculator shows an error for $\ln 0$ and $\ln(-3)$.

Mistake: Pressing LOG instead of LN on the calculator

Why: LOG is the common logarithm with base 10, LN is the natural logarithm with base $e$. They give different results.

Correct: $\log 10 = 1$, but $\ln 10 \approx 2.303$. Use LN whenever the base is $e$.

Mistake: Solving $e^x = 20$ by dividing: $x = \frac{20}{e}$

Why: $x$ is an exponent, not a factor. Dividing by $e$ does not undo raising $e$ to a power.

Correct: Undo $e^x$ with $\ln$: $x = \ln 20 \approx 2.996$.

Biology: When Will the Culture Reach a Target?

A bacteria culture with plenty of food grows continuously as $N = N_0 e^{rt}$. To find WHEN it reaches a size, you need $\ln$ to bring $t$ down from the exponent.

With $N = 1000e^{0.3t}$, the culture reaches 3000 when $e^{0.3t} = 3$, so $0.3t = \ln 3$ and $t = \frac{\ln 3}{0.3} \approx 3.66$ hours.

Finance: Doubling Time with Continuous Interest

Banks compare savings offers with continuous compounding, $A = Pe^{rt}$. The time it takes to double money is $t = \frac{\ln 2}{r}$.

At 7% compounded continuously, money doubles after $\frac{\ln 2}{0.07} \approx 9.90$ years. Since $\ln 2 \approx 0.693$, bankers use the "rule of 70": 70 divided by the rate in percent.

Medicine: How Long a Drug Stays in the Blood

Many medicines leave the body continuously, so the amount in the blood follows $C = C_0 e^{-kt}$. Doctors use $\ln$ to work out when the next dose is due.

A drug with $k = 0.25$ per hour has a half-life of $\frac{\ln 2}{0.25} \approx 2.77$ hours: after that time, half of it is left.

Q: Why is it called the natural logarithm?

A: Because its base $e$ comes up naturally: in continuous growth, in continuous compounding and in calculus, where $e^x$ and $\ln x$ have the simplest rates of change. The "n" in $\ln$ stands for the Latin "logarithmus naturalis".

Q: Is ln the same as log?

A: No. $\log x$ (common logarithm) has base 10 and $\ln x$ has base $e$. For example, $\log 100 = 2$ but $\ln 100 \approx 4.605$. Both answer the same kind of question, only the base is different.

Q: Why does my calculator give an error for ln(0)?

A: Because $e^y$ is always positive: there is no power of $e$ that equals $0$ or a negative number. The graph of $y = \ln x$ only lives to the right of the $y$-axis, which is a vertical asymptote.

Q: How are the graphs of $y = e^x$ and $y = \ln x$ related?

A: They are mirror images of each other in the line $y = x$, because they are inverse functions. The point $(0, 1)$ on $y = e^x$ matches the point $(1, 0)$ on $y = \ln x$, and $(1, e)$ matches $(e, 1)$.