Review of the logarithmic function base e

The inverse identities for e^x and ln x, the change-of-base formula, exponential equations that reduce to quadratics, and logarithmic scales.

Inverse identities

$$e^{\ln x} = x \ \ (x > 0) \qquad \ln e^x = x \ \ (\text{all } x)$$

Example 1●

Simplify (a) $e^{3\ln 2}$, (b) $\ln e^5$ and (c) $e^{-\ln 4}$.

$e^{3\ln 2} = e^{\ln 2^3} = 8$

$\ln e^5 = 5$

$e^{-\ln 4} = e^{\ln \frac14} = \tfrac14$

Change of base

Change-of-base formula

$$\log_a x = \frac{\ln x}{\ln a}$$

Example 2●

(a) Find $\log_2 7$ to 3 decimal places. (b) Solve $2^x = 5$ to 3 decimal places.

$\log_2 7 = \dfrac{\ln 7}{\ln 2} \approx 2.807$

$x = \log_2 5 = \dfrac{\ln 5}{\ln 2} \approx 2.322$

Equations that reduce to quadratics

Example 3●●

Solve $e^{2x} - 5e^x + 6 = 0$.

Let $u = e^x$, so $e^{2x} = u^2$: $u^2 - 5u + 6 = 0$.

Substitute

$(u - 2)(u - 3) = 0$, so $e^x = 2$ or $e^x = 3$.

$x = \ln 2$ or $x = \ln 3$.

Logarithmic scales

On a base-10 logarithmic scale, the numbers $1, 10, 100, 1000, \dots$ are equally spaced: each step of $1$ along the scale is a multiplication by $10$. Such scales are used when quantities range over many powers of ten, such as sound intensity (decibels), acidity (pH) and earthquake strength.

Example 4: decibels●●

Sound level in decibels is $L = 10\log_{10}\dfrac{I}{I_0}$. How many times more intense is an $80$ dB sound than a $50$ dB sound?

Every $10$ dB multiplies the intensity by $10$. The difference is $30$ dB, so the factor is $10^3 = 1000$.

Flashcards

$e^{ln x}$

$x$, for $x > 0$

$log_a x$ in base $e$

$\dfrac{\ln x}{\ln a}$

$e^{2x}$ in terms of $u = e^x$

$u^2$

Practice

Simplify $e^{3\ln 2}$.
Simplify $\ln e^5$.
Simplify $e^{-\ln 4}$.
Find $\log_2 7$ to 3 decimal places.
Solve $2^x = 5$ to 3 decimal places.
The solutions of $e^{2x} - 5e^x + 6 = 0$ are:
Solve $e^{2x} - e^x - 2 = 0$. Type $\ln 2$ as ln2.
How many times more intense is an $80$ dB sound than a $50$ dB sound?
On a base-10 logarithmic scale, the numbers $1, 10, 100, 1000$ are: