Converting between exponential and logarithmic form

How do write the exponential equation $2^x=8$ in logarithmic form?

How do you write the logarithmic equation $\log_2 8 = x$ in exponential form?

In case you didn’t notice, these two statements are mathematically equivalent, since the solution to both equals $3$, but what just happened? In the exponential equation, the base ‘$2$’ became the logarithm’s base, while the exponent $x$ got isolated in the logarithmic equation. This relationship is summarized below, and you’ll need to know this for future reference.

For the next two examples, remember the following:

Convert the following equations to logarithmic form:

 53=125 100=102

 log5125=3 log10100=2By convention, when the base of the log is 10, we don’t write it in as it’s assumed: ∴ log100=2

Convert the following equations to exponential form:

 log82=x log51125=x

Solutions:

 8x=223x=223x=213x=1x=13 5x=11255x=1535x=5-3x=-3

Always remember that the base of the exponent always becomes the base of the log, no matter what. Interesting though, there are two exceptions listed below:

1. If the base is 10, you don’t write the base 10 as it’s assumed to be 10 when not written. These logs are referred to as common logs. A history note: common logarithms are also called Briggs’ logarithms, after Henry Briggs (1561–1630).
2. If the base is e (euler’s number), rather than writing loge, it gets its own symbol as ln (read as “lawn”).

Convert the exponential equations into logs:

 100=102 e3=x

 log10100=2log100=2 logex=3lnx=3

Logarithms and Significant Figures

When a log is evaluated, let’s say:

log17.56=1.244524512

The answer 1.244524512 consists of the whole number 1, called the characteristic, and the positive decimal number 0.2445…, called the mantissa. The mantissa is rounded based on the number of digits being logged; notice that 17.56 consists of 4 numbers, hence we round to four significant figures. In other words, the characteristic is never accounted for a significant figure.

∴ log(17.56)=1.2445

Logarithm Restrictions

Try calculating log(-2) on your calculator. Doing so will result in a error. To explain why, set log(-2) equal to x, and then convert it into an exponential equation:

y=log(-2)10x=-2

The base is positive and is being raised to ‘x’. No matter what power you place into y, you will never get a negative output. This suggests that when you’re finding your restrictions for logarithmic functions, also remember that what you place into the log function must be greater than zero:

y=logx⏟x>0

Solving Basic Logarithmic Equations

Given what you’ve just learned about converting between logarithmic and exponential equations, let’s try solving the examples presented below:

Question 1

The temperature T of a bronze casting initially at 1250º F decreases exponentially with time. Its temperature after 5.00 s can be found from the equation:

logT1250=-1.55Evaluate ‘T’.

Solution

Start by changing the equation to exponential form:

10-1.55=y1250Now solve for ‘y’:

1250·10-1.55=yy=35.2The temperature after 5 seconds is 35.2 ºF.

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Question 2

A certain pendulum, after being released from a height of 4.00 cm, reaches a height y after 3.00 seconds, where y can be found from:

lny4.00=-0.50Evaluate ‘y’.

Solution

Start by changing the equation to exponential form:

logey4.00=-0.50e-0.50=y4.004.00·e-0.50=yy=2.43The height after 3 seconds is 2.43 cm.

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