### Logarithms Part I

Intro to logarithms

\begin{aligned} y &= a^x \\ log_{a} y &= log_{a} a^x \\ log_{a} y &= x\\ \end{aligned}

y, $\ a^x$ is known as the argument
a is known as the base
x is known as the exponent or the power

It is important to recognize that the base of the argument (a) equals to the base of the log (a)

Example 1

\begin{aligned} y &= a^x \\ 3 &= 10^x\\ log_{10} 3 &= log_{10} 10^x \\ log_{10} 3 &= x\\ \end{aligned}

• log base 10 is commonly used as it can be evaluated using a calculator.

• In most cases, it is abbreviated as lg. $\ log_{10}$ is equivalent to $\ lg$.

• Another number that is commonly used is 2.718, commonly known as e.

• log base e (2.718) can also be evaluated using a calculator.

• $\ log_{e}$ is equivalent to $\ ln$.

Example 2

\begin{aligned} 5 &= e^x\\ log_{e} 5 &= log_{e} e^x \\ log_{e} 5 &= x\\ x &= 1.61 \quad \text{to 3 s.f.} \end{aligned}

The next example will show you how Taking log on both sides can help you to answer questions

Example 3

Solve $10^{2x} - 2 ( 10^{x} ) -3 =0$

We start by using the Substitution $A = 10^x$

\begin{aligned} A^2 - 2A - 3 = 0 \\ (A - 3) ( A + 1) = 0 \\ A = 3 \qquad A = -1 \\ \text{Recall that } A = 10^x \\ 10^x = 3 \quad 10^x = -1 \\ 10^x \neq -1 \\ \text{hence we reject } 10^x = -1 \\ 10^x = 3 \\ \text{ taking log on both sides } \\ \log_{10} 10^x = \log_{10} 3 \\ \text{ use calculator to evaluate } \log_{10} 3 \\ x = 0.477 \\ \end{aligned}

Logarithms Part II will illustrate how we can use the "Taking log on both sides" technique in practical situations