The Natural Base e: Practice

The constant $e$ is the number that $\left(1 + \frac{1}{n}\right)^{n}$ closes in on as $n$ grows without end.

$e = 2.718281828459045\ldots \approx 2.71828$

The value $3.14159$ is $\pi$, a different irrational constant, and $1.61803$ is the golden ratio.

The natural logarithm is defined as the logarithm whose base is $e$.

$\ln x = \log_{e} x \quad (x > 0)$

The notation $\log_{10} x$ is the common logarithm (base $10$), and $e^{x}$ is the exponential function, which is the inverse of $\ln x$, not the same as it.

By definition, $\ln 7$ is the exponent you must put on $e$ to get $7$. Raising $e$ to that exponent therefore returns $7$.

$e^{\ln x} = x \quad (x > 0) \;\Longrightarrow\; e^{\ln 7} = 7$

The exponential and the natural logarithm undo each other.

The natural logarithm asks for the exponent on $e$, and here the exponent is written right there.

$\ln\left(e^{x}\right) = x \;\Longrightarrow\; \ln\left(e^{5}\right) = 5$

You can also get this from the power rule: $\ln\left(e^{5}\right) = 5\ln e = 5 \cdot 1 = 5$.

Continuous compounding is what the discrete formula closes in on as the number of compoundings per year grows without end.

$A = P\left(1 + \frac{r}{n}\right)^{nt} \;\longrightarrow\; A = Pe^{rt}$

The telltale sign is that no $n$ appears anywhere in the continuous formula: the question of how often to compound has stopped mattering.

The rate in the model is a decimal, not a percentage, because a percent means "per hundred".

$3\% = \frac{3}{100} = 0.03$

Using $r = 3$ would mean $300\%$ growth per year, which after one year multiplies the money by $e^{3} \approx 20.09$ instead of by $e^{0.03} \approx 1.03$.

A logarithm is an exponent. Since $\ln = \log_{e}$, the value $\ln x$ is the exponent that turns $e$ into $x$.

$\ln x = y \quad \text{exactly when} \quad e^{y} = x$

The option $10^{y} = x$ describes the common logarithm $\log x$, a different base.

Square the value of $e$ rather than doubling it, because the exponent multiplies the base by itself.

$e^{2} \approx (2.718282)^{2} \approx 7.389056$

The trap is $5.44$, which is $2e$, not $e^{2}$.

The natural logarithm obeys the product rule, because it is an ordinary logarithm with base $e$.

$\ln 3 + \ln 5 = \ln(3 \cdot 5) = \ln 15$

A sum of logarithms becomes the logarithm of a product, so the inputs multiply. It does not become $\ln 8$, which would come from adding the inputs.

Use $A = Pe^{rt}$ with $P = 400$, $r = 0.05$ (the decimal form of $5\%$) and $t = 2$. The exponent is the product $rt$.

$A = 400e^{(0.05)(2)} = 400e^{0.10}$

The rate and the time multiply inside the exponent; they are never added, and the percent is never used raw.

The natural logarithm is the logarithm with base $e \approx 2.71828$.

$\ln x = \log_{e} x$

The key marked $\log$ is the common logarithm, base $10$. Base $1$ is impossible for a logarithm, since $1$ raised to any power is $1$.

The rule $\ln\left(e^{x}\right) = x$ holds for every real $x$, including negative ones.

$\ln\left(e^{-3}\right) = -3$

The value $\frac{1}{e^{3}}$ is what $e^{-3}$ itself equals, but the question asks for its natural logarithm, which is the exponent.