How to Determine an Unknown Exponent

If you see the expressions 32 and 53, you might announce with a thrive that these mean “ three squared ” and “ five cubed, ” and be able to go about finding equivalent numbers without 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.

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.

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.

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