# Solving exponential and logarithmic equations kws006/Precalculus/3.5_Solving... Solving exponential

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Elementary Functions Part 3, Exponential Functions & Logarithms

Lecture 3.5a, Solving Equations With Logarithms

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 16

Solving exponential and logarithmic equations

We explore some results involving exponential equations and logarithms.

In this presentation we concentrate on using logarithms to solve exponential equations. As a general principle, whenever we seek the value of a variable in an equation:

If the variable appears as an exponent, we should think about using logarithms.

Smith (SHSU) Elementary Functions 2013 2 / 16

Solving exponential and logarithmic equations

Here is a set of sample problems. (The first four problems are from “Example 2” in Dr. Paul’s online math notes on logarithms at Lamar University.)

Example Solve the following exponential equations for x.

1 7x = 9

2 24y+1 − 3y = 0. 3 et+6 = 2.

4 5e2z+4 − 8 = 0 5 105x−8 = 8.

Solutions. In each case, since we are solving for a variable in the exponent, we may take a logarithm of both sides of the equation. In most cases, the base of the logarithm is irrelevant but in problems (3) and (4) we might as well use base e; in problem (5) we take the logarithm base 10.

Smith (SHSU) Elementary Functions 2013 3 / 16

Solving exponential and logarithmic equations

Example Solve the following exponential equations for x.

1 7x = 9

Solutions.

1 Apply ln() to both sides of 7x = 9 to obtain

ln(7x) = ln(9)

and so (by the “exponent” property of logs)

x ln(7) = ln(9)

and so x = ln 9ln 7 .

Smith (SHSU) Elementary Functions 2013 4 / 16

http://tutorial.math.lamar.edu/Classes/Alg/SolveExpEqns.aspx

Solving exponential and logarithmic equations

Example Solve the following exponential equations for x.

2 24y+1 − 3y = 0.

Solutions.

2 Rewrite 24y+1 − 3y = 0 as 24y+1 = 3y. Apply ln() to both sides of the equation to obtain

ln(24y+1) = ln(3y)

and pull out the exponents

(4y + 1) ln 2 = y ln 3.

Isolate y. 4y ln 2− y ln 3 = − ln 2.

Factor out y: y(4 ln 2− ln 3) = − ln 2.

Solve for y by dividing both sides by the constant 4 ln 2− ln 3 to get

y = − ln 24 ln 2−ln 3 = ln 2

ln 3−ln 16 .Smith (SHSU) Elementary Functions 2013 5 / 16

Solving exponential and logarithmic equations

Example Solve the following exponential equations for x.

3 et+6 = 2. 4 5e2z+4 − 8 = 0

Solutions.

3 Apply ln() to both sides of et+6 = 2 to obtain t+ 6 = ln 2 so

t = ln 2− 6. 4 Apply ln() to both sides of 5e2z+4 = 8 to obtain

ln 5e2z+4 = ln 8

Use the “multiplication” property of logs to rewrite this as

ln 5 + ln e2z+4 = ln 8

and so ln 5 + 2z + 4 = ln 8

and so 2z = ln 8− ln 5− 4.

z = ln 8−ln 5−42 . Smith (SHSU) Elementary Functions 2013 6 / 16

Solving exponential and logarithmic equations

Example Solve the following exponential equations for x.

5 105x−8 = 8.

Solutions.

5 Apply log() to both sides of 105x−8 = 8 to obtain 5x− 8 = log 8 and so

x = log 8+85 .

(Note that here we are using 10 as the base of our logarithm in this problem.)

Smith (SHSU) Elementary Functions 2013 7 / 16

Solving exponential and logarithmic equations

Some more worked problems. Here are some problems off of an old exam: Solve for x in the following equations, finding the exact value of x. Then use your calculator to approximate the value of x to four decimal places.

1 2x = 17 2 2x = 3x+1

Solution.

1 x = log2(17) = ln 17 ln 2 ≈ 4.08746284.

2 Take the natural log of both sides of the equation 2x = 3x+1 to obtain

ln(2x) = ln(3x+1)

Use the exponent property to pull down the exponents and then isolate x:

x ln 2 = (x+ 1) ln 3

x ln 2 = x ln 3 + ln 3.

x ln 2− x ln 3 = ln 3 x(ln 2− ln 3) = ln 3.

x = ln 3ln 2−ln 3 ≈ −2.70951129 Smith (SHSU) Elementary Functions 2013 8 / 16

Solving exponential and logarithmic equations

Sometimes our equation explicitly involves logarithms and we need to use properties of logarithms to get the problem into the correct form where we can easily solve it.

Here is an example: Solve the equation

log3(2x 2 − 8)− log3(x− 2) = 4.

Solution. We solve log3(2x 2 − 8)− log3(x− 2) = 4 by using our quotient

property to rewrite the lefthand side as log3 2x2−8 x−2 . We may factor 2x

2 − 8 as 2(x− 2)(x+ 2) and so (as long as x 6= 2) simplify 2x2−8 x−2 = 2(x+ 2) = 2x+ 4. So out equation simplifies to

log3(2x+ 4) = 4.

We rewrite this log equation into exponential form, removing the logarithm from the problem.

2x+ 4 = 34 = 81

We can (easily!) solve nice linear equations like 2x+ 4 = 81 and get

x = 772 .

Smith (SHSU) Elementary Functions 2013 9 / 16

Exponential Functions

In the next presentation we continue to practice applications of logarithms.

(END)

Smith (SHSU) Elementary Functions 2013 10 / 16

Elementary Functions Part 3, Exponential Functions & Logarithms

Lecture 3.5b, Solving Equations With Logarithms, continued

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 11 / 16

Solving exponential and logarithmic equations

Modern scientific computations sometimes involve large numbers (such as the number of atoms in the galaxy or the number of seconds in the age of the universe.) Some numbers are so large it is difficult to even figure out the number of digits. Here is an example. How many decimal digits are there in the number 23003100? We can answer this question by computing the logarithm of the number base 10 and correctly interpreting the result.

If N is a positive integer then the number of decimal digits in 10N is N + 1 since 10N is written as a one followed by N zeroes. Since the logarithm base 10 of 10N is just N , we see that the number of decimal digits in a number is one more than the floor of the logarithm base ten.

Smith (SHSU) Elementary Functions 2013 12 / 16

Solving exponential and logarithmic equations

For example,

log(23003100) = log(2300) + log(3100) = 300 log 2 + 100 log 3.

We can approximate log 2 ≈ 0.30103 and log 3 ≈ 0.47712 to write

300 log 2+100 log 3 ≈ 300(0.30103)+100(0.47712) = 90.309+47.712 = 138.021.

This tells us that 23003100 ≈ 10138.021.

Note that 101 = 10 has two decimal digits, 102 = 100 has three decimal digits and in general, if we want the decimal digits of an expression of the form 10x, we need to round up. So 10138.021 has 139 decimal digits.

We can say more. We can approximate 10138.021 = 100.021 × 10138 ≈ 1.05× 10138. So 23003100 begins “105...” and continues with another 136 digits!!

Smith (SHSU) Elementary Functions 2013 13 / 16

Solving exponential and logarithmic equations

The largest known prime number. There are an infinite number of primes. This was first proven by Euclid around 300 BC! However, we only know, at this time, a finite number of prime numbers.

Large prime numbers play a role in computer science and digital security. As of June 2013, the largest known prime number (according to Wikipedia) is 257,885,161 − 1. This is a big number! Suppose I want write out this big prime number. How many decimal digits would it take? Solution.

log10(2 57,885,161) = 57885161 · log10(2) = 57885161 ln 2ln 10 ≈ 57885161(0.30103) ≈ 17425169.76484.

This means that

257,885,161 ≈ 1017425169.76484 = (100.76484)(1017425169) ≈ 5.81887× 1017425169.

In other words, 257,885,161 begins with a 5 and is followed by another 17425169 digits, so it has 17,425,170 digits! That’s over 17 million digits!Smith (SHSU) Elementary Functions 2013 14 / 16

Exponential Functions

I certainly don’t want to try to write out the 17,425,170 digits of 257,885,161 − 1.

Indeed, it might be hard to get a computer system to write that out, although you could give WolframAlpha a try.

In the solution to this problem on prime numbers, I calculated the number of digits in 257,885,161. But the prime number we were after is really 257,885,161 − 1. Is it obvious that subtracting 1 from 257,885,161 won’t change the number of digits?

Smith (SHSU) Elementary Functions 2013 15 / 16

Exponential Functions

In the next presentation we continue to practice applications of logarithms.

(END)

Smith (SHSU) Elementary Functions 2013 16 / 16

http://en.wikipedia.org/wiki/Largest_known_prime_number

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