I need help with understanding Laws of Logarithms and exponential functions

Question

Mayme · Accepted Answer

Step 1
Laws of logarithms and exponential functions
first let’s see what are laws of logarithms and exponential functions
Step 2
Laws of exponents:

a^{1} = a

This is a natural sequence of exponent

a^{0} = 1

This is a natural sequence of exponent

a^{- 1} = \frac{1}{a}

This is also a natural sequence of exponent

a^{m} a^{n} = a^{m + n}

Here we are multiplying ‘a’m times and then n times for a total of m+n times

\frac{a^{m}}{a^{n}} = a^{m - n}

Here we are dividing the a which is multiplied m times and n times. Reducing it I.e. m-n times

{(a^{m})}^{n} = a^{m} n

first multiply a m times and then you have to do that n times I.e. mn times

{(a b)}^{m} = a^{m} b^{m}

{(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}

Step 3
Laws or properties of logarithms

\log_{a} (x y) = \log_{a} x + \log_{a} y

The log of multiplication is sun of the logs

\log_{a} (\frac{x}{y}) = \log_{a} x - \log_{a} y

The log of division is the difference of logs

\log_{a} (x^{r}) = r \log_{a} x

The log of x with an exponent r is r times the log of x
The natural logarithm

\log_{e} x

which is written as

\ln (x)

The common logarithm is with base 10 which is written as

\log_{10} x (or) \log (x)

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