Example  Evaluate each of the following logarithms.

(a) log1000

 (b) log 1/100

 (c) ln1/e

 (d) ln √e

(e) log³⁴ 34

(f) log₈ 1

Solution

In order to do the first four evaluations we only have to remember what the notation for these are & what base is implied by the notation. The last two evaluations are to show some properties of all logarithms which we'll be looking at eventually.

 (a) log1000 = 3 since 10³  = 1000 .

 (b) log  1/100 = -2 since 10^-2  =  1/10²  =  1/100

 (c) ln 1/e = -1 since e^-1 = 1 .

 (d) ln √e = 1 /2 since e ^1/2  = √e .

 Notice that along with this one we are actually just acknowledging variation of notation from fractional exponent in radical form.

 (e) log₃₄  34 = 1 since 34¹ = 34 .Notice that this one will work regardless of the base that we're using.

 (f) log₈ 1 =0 since 8⁰ =1 Again, note that the base which we're using here won't alter the answer.

Thus, while evaluating logarithms all that we're actually asking is what exponent did we put onto the base to obtain the number in the logarithm.

Now, before we get into some of the properties of logarithms let's first do a couple of quick graphs.