**exponents**, the numbers represented by the superscripts to the upper berth correct above. These numbers in this subject are 9 and 125. But what if, rather of, say, a simple exponential routine such as yttrium = x 3, you alternatively have to solve an equation like y = 3x. here ten, the dependent variable, appears as an exponent. Is there a way to pull that variable down from its perch to more easily deal with it mathematically ? In fact there is, and the answer lies in the natural complement of exponents, which are fun and helpful quantities known as

**logarithms**.

Reading: How to Determine an Unknown Exponent

Table of Contents

## What Are Exponents?

An **exponent**, besides called a office, is a compressed way of expressing perennial multiplications of a act by itself. 45 = 4 × 4 × 4 × 4 × 4 = 1,024.

- Any number raised to the power of 1 keeps the same value; any number with an exponent of 0 is equal to 1. For example, 721 = 72; 720 = 1.

Exponents can be damaging, producing the relationship **x** **−n** **= 1/(xn)**. They can besides be expressed as fractions, for example, 2 ( 5/3 ). If expressed as fractions, both the numerator and denominator must be integer numbers.

## What Are Logarithms?

Logarithms, or “ logs, ” can be regarded as exponents expressed as something other than a power. That credibly does n’t help much, therefore possibly an exercise or two will. In the expression **103 = 1,000**, the count 10 is the **base**, and it is being raised to the third power ( or power of three ). You can express this as, “ the base of 10 raised to the third power equals 1,000. ” An example of a logarithm is **log10(1,000) = 3**. note that the numbers and their relationships to each other are the lapp as in the previous example, but they have been moved about. In words, this means, “ the log base 10 of 1,000 equals 3. ” The quantity on the proper is the power that the base of 10 has to be raised to in rate to equal the **argument**, or input signal of the log, the respect in parentheses ( in this lawsuit 1,000 ). This value has to be cocksure, because the free-base — which can be a act early than 10, but is assumed to be 10 when omitted, e.g., “ log 4 ” — is besides always convinced.

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## Helpful Logarithm Rules

So how can you work easily between logs and exponents ? A few rules about the behavior of logs can get your started on exponent problems. log_ { b } ( xy ) = log_ { b } { ten } + log_ { barn } y log_ { bacillus } ( \dfrac { x } { y } ) = log_ { b } { adam } \text { − } log_ { bacillus } y log_ { barn } ( x^A ) = A⋅log_ { bacillus } ( ten ) log_ { b-complex vitamin } ( \dfrac { 1 } { yttrium } ) = −log_ { b-complex vitamin } ( y )

## Solving for an Exponent

With the above data, you ‘re ready to try solving for an advocate in an equation. example : If 50 = 4x, what is x ? If you take the logarithm to the base 10 of each side and neglect denotative identification of the base, this becomes log 50 = log 4x. From the box above, you know that log 4x = x log 4. This leaves you with log 50 = x log 4, or x = ( log 50 ) / ( log 4 ). Using your calculator or electronic device of choice, you find that the solution is ( 1.689/0.602 ) = **2.82**.

## Solving Exponential Equations With e

The lapp rules apply when the base is vitamin e, the alleged **natural logarithm**, which has a value of about 2.7183. You should have a button for this on your calculator as well. This respect gets its own notation, besides : logex is written just “ ln x. ”

- The function y = ex i, with e not a variable but a constant with this value, is the only function with a slope equal to its own height for all x and y.
- Just as log1010x = x, ln ex = x for all x.

**Example:** Solve the equality 16 = e2.7x. As above, ln 16 = ln e2.7x = 2.7x. ln 16 = 2.77 = 2.7x, thus ten = 2/77/2.7 = **1.03.**