Exponents & Logarithms

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Logarithm algebra

Discussion

Jimmy has a piece of paper and he notices that when he folds it in half, its thickness doubles. He measures the initial thickness of the paper as 0.1mm. He wonders how many times he would have to fold the paper to reach the moon, which he knows is 384000km from Earth.

How many times does Jimmy need to fold the piece of paper to reach the moon?

We fold the paper n times, so its thickness becomes

tn=0.1mm×2n.

Since 0.1mm=10−7km,

tn=10−7×2n(km).

We need tn≥384000km. By guessing:

n=10:t10=10−7×210=10−7×1024≈1.02×10−4km

We see 10 folds is far too few. So we try:

n=100:t100=10−7×2100=10−7×1.27×1030≈1.27×1023km

Clearly, we overadjusted, as 100 would be far too many folds. Now that we know Jimmy should make between 10 and 100folds we make successive guesses:

n=20:t20=10−7×220=10−7×1.05×106≈0.105km,n=30:t40=10−7×230=10−7×1.07×1012≈107kmn=40:t40=10−7×240=10−7×1.10×1012≈1.10×105km.

At n=40, t40≈110000 km, still about a third of Jimmy's desired thickness of 384000 km. One more fold doubles it, which would not be enough, but two more folds will quadruple the thickness to about 440000km:

t42≈22×1.10×105=4.40×105km,

which exceeds 384 000 km. Hence Jimmy must fold the paper 42 times.

As you can see, we do not yet have a good way to solve problems in which we need solve exponentials with very large powers. In this lesson, we will learn a more systemic way to solve problems like Jimmy's.

Basically, Jimmy's question can be summed up mathematically as, "For what n does 2n=3.84×1012?"

This question exposes that we do not have a good way for solving exponential equations or inequalities for numbers that cannot be simply rewritten as an exponent of the desired base.

What we need for this is an inverse operation of the exponent that can tell us what exponent a given base needs to be raised to for the desired answer. This operation is called a logarithm.

Definition of the logarithm

Logarithms are a mathematical tool for asking "what power of a given base gives a specific value". We write this as

logab=x⇔ax=b.

Here, a is called the base, and must be positive. b must also be positive. The value of x, however, can be any real number.

Checkpoint

Evaluate log636.

Select the correct option

Exercise

Evaluate log3(271).

Select the correct option

Certain log bases are so common that they have shorthand notation

log base 10

In science and mathematics it is so common to use log10 that we can simply write the shorthand log to indicate log10.

For example, log(0.001)=−3 since 10−3=0.001.

Exercise

Evaluate log0.1.

Select the correct option

Natural logarithm

Another special logarithm is the one in base e. We call it the natural logarithm due to fundamental importance of e across mathematics.

loge is the same as ln

For example, ln(e3)=3.

Exercise

Evaluate ln(3√e4).

Select the correct option

Evaluating logs

Some logarithms can be evaluated by hand using the fact that

logab=x⇒ax=b

If a and b are not powers of the same base, the log cannot be computed by hand. But we can use a calculator to evaluate them approximately.

log35≈1.46

On a TI-84, MATH>logBASE(can be used for bases other than 10 and e.

Exercise

Evaluate each of the following logs using a calculator:

log(19)
ln(4)
log76

Select the correct option

Discussion

Now that we understand log notation, we can explore algebraic properties of logs.

Find x such that log(x)=log(100)+log(1000). What do you notice about x?

Sum and difference of logs

The sum of logs with the same base is the log of the products:

logax+logay=loga(xy)📖

We have a similar rule for the difference of logs:

logax−logay=loga(yx)📖

Checkpoint

Evaluate log6(3)+log6(2).

Select the correct option

Discussion

Solve xlog39=log381. What do you notice?

Log power rule

loga(xm)=mlogax📖

Exercise

Write log(27ab29a3b) as a sum / difference of logs.

Select the correct option

Discussion

Evaluate log10log1000. What do you notice about your answer?

Log change of base

We can change the base of a logarithm using the law

logax=logbalogbx📖

for any choice of positive b.

Exercise

Using the log change of base rule, evaluate log8127.

Select the correct option

Exercise

Write as a single logarithm log83log86.

Select the correct option

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Discussion

Now that we know how to use logs, can you find a more straightforward way to solve Jimmy's problem: how many times do we need to fold a 0.1mm to reach the moon 384000km away?

Without logarithms, we haf to keep doubling 0.1 mm (or 0.000 000 1 km!) by trial and error. A log solves it directly.

First convert the paper’s initial thickness to kilometres:

0.1mm=0.1×10−3m=0.1×10−6km=1.0×10−7km

After n folds the thickness is

1.0×10−7×2n km,

so we require

1.0×10−7×2n≥384000⟹2n≥1.0×10−7384000=3.84×1012.

Taking log (base 10) on a TI-84:

n≥log(2)log(3.84E12)≈41.847

(Using change of base, we see this is equivalent to log2(3.84×1012) but easier to type on a calculator with a log (base 10) button! You could do the same thing using ln.)

Since n must be an integer and we need at least that thickness,

n=42

folds.

You just used a logarithm to solve an exponential equation! Notice, logarithms are a helpful tool when equating exponents is not possible.

Using logs to solve exponential equations

Logarithms can be used to solve exponential equations:

ax=b⇔x=logab.

Checkpoint

Evaluate 3x=74.

Select the correct option

Exercise

Solve the equation 4x+2=20, giving your answer as a single log.

Select the correct option

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