Laws of Logarithms

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Laws of logarithms allow you to simplify and manipulate expressions involving logarithms
- The laws of logarithms are equivalent to the laws of indices
The laws you need to know are, given $a, x, y > 0$ :
- $\log_{a} x y = \log_{a} x + \log_{a} y$
  - This relates to $a^{x} \times a^{y} = a^{x + y}$
- $\log_{a} \frac{x}{y} = \log_{a} x {- \log}_{a} y$
  - This relates to $a^{x} \div a^{y} = a^{x - y}$
- $\log_{a} x^{m} = m \log_{a} x$
  - This relates to $(a^{x})^{y} = a^{x y}$
These laws are in the formula booklet so you do not need to remember them
- You must make sure you know how to use them

Laws of Logarithms Notes fig2

Given $a > 0, a \neq 1$
- $\log_{a} 1 = 0$
  - This is equivalent to $a^{0} = 1$
If we substitute b for a into the given identity in the formula booklet
- $a^{x} = b \Leftrightarrow \log_{a} b = x$ where $a > 0, b > 0, a \neq 1$
- $a^{x} = a \Leftrightarrow \log_{a} a = x$ gives $a^{1} = a \Leftrightarrow \log_{a} a = 1$
  - This is an important and useful result
Substituting this into the third law gives the result
- $\log_{a} a^{k} = k$
Taking the inverse of its operation gives the result
- $a^{\log_{a} x} = x$
From the third law we can also conclude that
- $\log_{a} \frac{1}{x} = - \log_{a} x$

Laws of Logarithms Notes fig3

These useful results are not in the formula booklet but can be deduced from the laws that are
Beware…
- … $\log_{a} (x + y) \neq \log_{a} x + \log_{a} y$
These results apply to $\ln x (\log_{e} x)$ too
- Two particularly useful results are
  - $\ln e^{x} = x$
  - $e^{\ln x} = x$
Laws of logarithms can be used to …
- simplify expressions
- solve logarithmic equations
- solve exponential equations

Remember to check whether your solutions are valid
- log (x+k) is only defined if x > -k
- You will lose marks if you forget to reject invalid solutions

Write the expression

2 \log 4 - \log 2

in the form

\log k

, where

k \in ℤ

aa-sl-1-2-2-laws-of-logs-we-solution-part-a

b) Hence, or otherwise, solve

2 \log 4 - \log 2 = - \log \frac{1}{x}

aa-sl-1-2-2-laws-of-logs-we-solution-part-b

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Change of Base

The laws of logarithms can only be used if the logs have the same base
- If a problem involves logarithms with different bases, you can change the base of the logarithm and then apply the laws of logarithms
Changing the base of a logarithm can be particularly useful if you need to evaluate a log problem without a calculator
- Choose the base such that you would know how to solve the problem from the equivalent exponent

$\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$

This is in the formula booklet
The value you choose for b does not matter, however if you do not have a calculator, you can choose b such that the problem will be possible to solve

Changing the base is a key skill which can help you with many different types of questions, make sure you are confident with it
- It is a particularly useful skill for examinations where a GDC is not permitted

By choosing a suitable value for b, use the change of base law to find the value of $\log_{8} 32$ without using a calculator.

aa-sl-1-2-2-change-of-base-we-solution-1