XXII Logarithms

Rules and Examples

$\log_4 16 = 2 \leftrightarrows 4^2 = 16$

$\log_b n = e \leftrightarrows b^e = n$

Given $b = 10$:

$n = 10^e /Rarr n$ is expressed as a function of $e$
$\log_10 n = e \Rarr e$ is expressed as a function of $n$

$\log_x 1 = 0 \Rarr$ always $0$ regardless of the base because $n^0 = 1$

$\log (10) = 1 \Rarr$ lack of subscript implies a base of $10$.

Notice how in $10^x = n$, the number of zeros in $n$ is equal to $x$: $\log (100) = 2$
$\log (1000) = 3$
$\log (1000000) = 6$

Likewise: $\log (0.1) = -1$
$\log (0.01) = -2$
$\log (0.001) = -3$
$\log (0.00001) = -5$

As a rule in $\log (x)$, $x>0$ has to be true.

Further rules

1. If a fraction is given to the log function, the result is always a negative number.
2. If the number given is greater than the base, the result is always a whole number.
3. If the base is greater than the given number, the result is always a fraction of a number < 1.

Change of Base Formula

$\log$a (b) = \frac{\logc (b)}{\log_c (a)}

Since calculators often only support base 10 and ln for log functions, this formula can be used to solve log with different bases:

$\log_4 (16) =$ ? but since $\frac{\log (16)}{\log (4)} = 2$

Properties of Logarithms

$\log (A) + \log (B) = \log (AB)$
$\log (A) - \log (B) = \log (\frac A B)$
$\log (A^2) = 2\log (A)$

Example 3 shows that logarithms can be used to algebraically find an unknown exponent in an equation:

$x = y^2 \Rarr \log x = 2 \log x$

$\log (x) + \log (y) - \log (z) = \log (\frac{xy}{z})$

Basically when adding and subtracting, positive logs move to the numerator and negative logs move to the denominator of a fraction.

Natural Log

Naturally arising logs in advanced mathematics and engineering are calculated to a base $e$ given by the sequence

$1 + \frac 1 1 + \frac 1 1.2 + \frac 1 1.23 + \frac 1 1.234 + \text{\textellipsis}$ to infinity

The sum of above sequence is $e$, which can be calculated to any degree of accuracy.
e.g. $e = 2.71828$

The natural logarithm $e$ is often abbreviated $\ln$.

So $\log_e 10 = \ln 10$

Simplification

$\ln 1 = 0$
$\ln e = 1$
$\ln e^5 = 5$

When a logarithm is taken as an exponent, and the base is equal to the logarithm's base, they cancel and only the input to the logarithm remains:

$e^{\ln7} = 7$

Logarithmic Equations

Considering the exponential form often helps in solving equations involving logarithms.

$\log_x 27 = 3$ solve for $x$

$x^3 = 27$
$x = \sqrt[3]{27}$
$x = 3$

When solving leads to multiple values for $x$, remember that the input $x$ in $\log (x)$ can never be $0$ or smaller.

Logarithm of a Root

Since logarithmic rules are derived from exponent laws it follows that

$\log \sqrt[4]{a} = \log a^{\frac 1 4}$
$= \frac 1 4 \log a$

Reciprocal

The logarithm of a number is equal to the negative of its reciprocal

$\log (\frac 1 9) = \log 1 - \log 9$
$= 0 - \log 9$
$= - \log 9$