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:

logBaseAnswer=Exponent⇔BaseExponent=Answer

Convert the following equations to logarithmic form:

53=125 100=102

Answers:

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:

- 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). - If the base is
*e*(euler’s number), rather than writing log, it gets its own symbol as_{e}**ln**(read as “lawn”).

Convert the exponential equations into logs:

100=102 e3=x

Answers:

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 1The 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’.

Start by changing the equation to exponential form:

10-1.55=y1250Now solve for ‘y’:

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

**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’.

Start by changing the equation to exponential form:

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