# Logarithmic Functions In Algebra

These are my notes on logarithmic functions in Algebra.

A logarithmic function is the inverse of an exponential function. It is used to model data that grows at an extremely slow rate. This is usually stated by \(g(x) = \log x\).

The graph of \( \log x \) is a continuous curve with the y-axis a vertical asymptote. The common logarithmic function is one-to-one and always increasing. Its domain is all positive real numbers with its range being all real numbers.

The logarithmic function outputs an exponent. The input to this function must be positive. Let’s do some problems out of my textbook.

**Example 1**

Simplify \(\log 1\)

1 is the same as \(10^0\)

So, \(\log 1 = 10^0\)

In logarithmic problems, we are looking for the exponent.

Our answer is \(0\)

**Example 2**

Simplify \(\log \frac{1}{1000}\)

\(\log \frac{1}{1000} = \log 10^{-3} = -3\)

**Example 3**

Simplify \(\log \sqrt{10}\)

\(\log \sqrt{10} = \log 10^{\frac{1}{2}} = \frac{1}{2}\)

**Example 4**

Simplify \(\log (-2)\)

The input to a log problem must be positive so this expression is undefined.

Logarithmic problems are based on the powers of ten. So, a logarithmic problem is asking what exponent with a base of ten gives the log number in question. So, \(\log 15 = 1.17\). This is:\[10^{1.17} = 15\].

A logarithm is an exponent.

\(\log 1000\) is the exponent that you raise 10 to to get 1000. That answer is 3.

Logarithms have inverse properties too. They are \(\log 10^{x} = x\) and \(10^{\log x} = x\). An exponential equation has a variable that occurs as an exponent in an expression. To solve the exponential equation \(10^x = k\), we take the common logarithm of each side and then apply the inverse property \(\log 10^x = x\) for any real number x.

**Example 5**

Simplify \(10^x = 5\)

Take the common logarithm

\(\log 10^x = \log 5\)

Apply the inverse property: \(\log 10^x = x\)

\(x = \log 5\)

**Example 6**

Simplify \(10^x = .001\)

Take the common logarithm of each side

\(\log 10^x = \log .001\)

Move your decimals over

\(x = \log 10^{-3} = -3\)

**Example 7**

Simplify \(10^x = 55\)

Take the common logarithm of each side

\(\log 10^x = \log 55\)

A log is the inverse to the power of ten

\(x = \log 55\)

Use a calculator or look at a log table

\(x = 1.74\)

**Example 8**

Simplify \(10^x = -1\)

This expression is undefined because the input must be positive and -1 is negative

**Example 9**

Simplify \(4(10^{3x}) = 244\)

The first thing that pops out at us is to get rid of the coefficient 4

Do this by dividing both sides by 4

\(10^{3x} = 61\)

Take the common logarithm of both sides

\(\log 10^{3x} = \log 61\)

The log and power of ten cancel out, this is the inverse property

\(3x = \log 61\)

Divide both sides by 3

\(x = \frac{\log 61}{3}\)

\(x = .59\)

Now, we are prepared to do logarithmic equations. A logarithmic equation contains logarithms. To solve logarithmic equations we exponentiate each side of the equation and then apply the inverse property \(10^{\log x} = x\).

**Example 10**

Simplify \(\log x = 2.5\)

Exponentiate each side

\(10^{\log x} = 10^{2.5}\)

Apply the inverse property \(\log 10^x = x\)

\(x = 10^{2.5}\)

\(x = 316.23\)

**Example 11**

Simplify \(\log x = 3\)

Exponentiate each side

\(10^{\log x} = 10^3\)

Apply the inverse property

\(x = 10^3\)

\(x = 1000\)

**Example 12**

Simplify \(\log x = -2 \)

This is the same as saying \(x = 10^{-2} = .01 \)

**Example 13**

Simplify \(\log x = 2.7 \)

This is the same as \(x = 10^{2.7} = 501.2 \)

**Example 14**

Solve \( 5 \log 2x = 16 \)

Divide each side by 5

\( \log 2x = 3.2 \)

Exponentiate each side

\( 10^{\log 2x} = 10^{3.2} \)

Use the inverse property

\(2x = 10^{3.2} \)

Divide each side by 2

\( x = \frac{10^{3.2}}{2} \)

\(x = 792.4 \)

Logarithms with any base other than one can be calculated. Any base logarithm has a domain of zero to infinity. This is the input (x) and it must be positive. The range of any base logarithm is negative infinity to infinity, so all real numbers. Remember that a logarithm is an exponent. So, we are just looking for the exponent of a certain base number when we are simplifying or solving logarithmic equations.

**Example 15**

Solve \(\log_6 6^{-1.3} \)

\( \log_a a^{x} = x = -1.3 \)

**Example 16**

Solve \( 5^{\log_5(x+8)} \)

\(a^{\log_a x = x} \) so \(5^{\log_5(x+8)} = (x+8) \)

**Example 17**

Solve \( \log_\frac{1}{2} (\frac{1}{2}^{45}) \)

\( \log_a a^{x} = x \) so, \( \log_\frac{1}{2} (\frac{1}{2})^{45} = 45 \)

The inverse function of \(f(x) = a^{x}\) is \( f^{-1} (x) = \log_a x \). The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.

**Example 18**

Evaluate \( \log_2 8 \)

Rewrite 8 as \(2^{k}\)

\(8 = 2^{3}\)

\( \log_2 8 = \log_2 2^{3} = 3 \)

**Example 19**

Evaluate \( \log_5 \frac{1}{25} \)

\( \log_5 \frac{1}{25} = \log_5 \frac{1}{5^{2}} = \log_5 5^{-2} = -2 \)

**Example 20**

Evaluate \( \log_7 49 \)

\( \log_7 49 = \log_7 7^{2} = 2 \)

**Example 21**

Evaluate \( (ln) e^{-7} \)

\( (ln) e^{-7} = \log_e e^{-7} = -7 \)

To solve the equation \(a^{x} = k\), take the base-a logarithm of each side.

**Example 22**

Solve \( 3^{x} = \frac{1}{27} \)

Take the base-3 logarithm of each side

\( \log_3 3^{x} = \log_3 \frac{1}{27} \)

Consider the properties of exponents

\( \log_3 3^{x} = \log_3 \frac{1}{3^{3}} \)

\( \log_3 3^{x} = \log_3 3^{-3} \)

Use the inverse property

\( x = -3 \)

**Example 23**

Solve \( e^{x} = 5 \)

Take the natural logarithm of each side

\((ln) e^{x} = (ln) 5 \)

\( x = (ln) 5 = 1.6 \)

**Example 24**

Solve \( \log_2 x = 5 \)

Exponentiate each side

\( 2^{\log_2 x} = 2^{5} \)

Use the inverse property

\( x = 2^{5} \)

\( x = 32 \)

**Example 25**

Solve \( \log_5 x = -2 \)

Exponentiate each side

\( 5^{\log_5 x} = 5^{-2} \)

Use the inverse property

\( x = 5^{-2} \)

\( x = \frac{1}{5^{2}} \)

\( x = \frac{1}{25} \)

**Example 26**

Solve \( (ln) x = 4.3 \)

\( e^{(ln) x} = e^{4.3} \)

\( x = e^{4.3} \)

\( x = 73.7 \)

**Example 27**

Solve \( 5e^{x} -8 = 37 \)

Add 8 to each side

\( 5e^{x} = 45 \)

Divide each side by 5

\( e^{x} = 9 \)

Take the natural logarithm of each side

\( (ln) e^{x} = (ln) 9 \)

Use the inverse property

\( x = (ln) 9 \)

Use calculator

\( x = 2.1 \)

**Example 28**

Solve \( 5 (ln) 2x +3 = 10 \)

\(5 (ln) 2x = 7 \)

Divide both sides by 5

\( (ln) 2x = \frac{7}{5} \)

Exponentiate each side

\( e^{(ln) 2x} = e^{\frac{7}{5}} \)

Use inverse property

\( 2x = e^{\frac{7}{5}} \)

Divide both sides by 2

\( x = \frac{e^{\frac{7}{5}}} {{2}} \)

Use calculator

\( x = 2.02 \)