Logarithmsthe inverse of exponential functions

The logarithmic functions help us work easily with very large or very small numbers….

While calculators have helped us do this, notice that the LOG and In buttons are STILL a part of the calculator and are still an important part of

higher mathematics.

Remember how we had to determine the x intercepts for some exponential graphs with

trial and error? Logs will deliver us from this!

I like to think of logs as exponents because of the following….

We must become masters of translating an exponential expression into logarithmic and

visa versa.

𝑪𝒐𝒏𝒔𝒊𝒅𝒆𝒓 :𝟒𝟑=𝟔𝟒

This is what we call “ exponential form”. Let’s change it to “logarithmic form”.

log4 64 = 3

Look closely how that translation went…

𝟒𝟑=𝟔𝟒log4 64 = 3

The exponent becomes what the log expression is equal to! See why I said logs are equal to exponents.The BASE in the exponential expression becomes the

BASE in the logarithmic expression.

You try it!

𝟐𝟓=𝟑𝟐log2 32 = 5

You need to translate in the opposite direction too!

𝟐−𝟑=𝟏𝟖

log2 = -3

Your calculator will ONLY calculate logs base 10. Log is called the “common log”. It is so common that when we are referring to log base 10 we don’t include the base.

log2 x

log5 14

log x

log 14

Your calculator will also calculate logs base e but it uses a different button. ( In) This log is called the natural log and must be used with e.

ln x

ln 45

Practice! EVALUATE:

log2 8It helps to write it into exponential form.

log2 8 = x

= 8So 2 to WHAT POWER results in 8?

Practice! EVALUATE:

log36 6It helps to write it into exponential form.

log36 6 = x

= 6So 36 to WHAT POWER results in 6?

Practice! EVALUATE:

log5 0.2It helps to write it into exponential form.

log5 0.2 = x

= 0.2So 5 to WHAT POWER results in 0.2?

Practice! EVALUATE:

125

125 = x

= 125

Practice! EVALUATE:

log 100It helps to write it into exponential form.

log 100 = x

= 100So 10 to WHAT POWER results in 100?

Practice! EVALUATE:

𝒆𝒍𝒏❑𝟐𝟎

it simplifies to 20!

Find the inverse of the function.

𝒚=𝒍𝒐𝒈𝟒𝒙Rewrite it in exponential form….

𝟒𝒚=𝒙THEN switch the x and y….

𝟒𝒙=𝒚

Find the inverse of the function.

-3)Rewrite it in exponential form….

𝟐𝒚=𝒙 −𝟑THEN switch the x and y….

𝟐𝒙=𝒚 −𝟑THEN solve for y

𝟐𝒙+𝟑=𝒚

Find the inverse of the function.

-3) + 2Move the 2 over….

Rewrite it into exponential form….

𝟐𝒚 −𝟐=𝒙 −𝟑Switch x and y and solve for y.

-3)