8. Solve for x: 3x = 4x - 1
This is an exponential eqution. Take logs of both sides
log 3x = log4x - 1
The rule which applies to this situation is the one which says that logarithms turn exponents into factors.
x log 3 = (x - 1) log 4
This is actually a linear equation. log 3 and log 4 are numbers. Follow the steps for solving linear equations. Remove parentheses.
x log 3 = x log4 - log 4
Transpose known terms to one side and unknown terms to the other. I would transpose the known terms to the left and the unknown terms to the right
log 4 = x log 4 - x log 3
One could transpose the terms to the other sides, but this way we have more positive terms. At this point the step to get all of our unknown terms together would be to factor out the common unknown factor from the two terms on the right.
log 4 = x (log 4 - log3)
At this point, our unknown appears on only one place, and only one thing is happening to it. It is being multiplied by log 4 = log 3. To get rid of multiplying by log 4 - log 3, divide by log 4 - log 3.
and this is the solution.
There is another way to solve this equation. The main idea in solving equations is to get all of the unknowns together so that there is only a single thing happning to it which we can reverse. We could rewrite the equation
3x = 4x - 1
If we multiply both sides by 4 and divide by 3x that will get all of the x's together on the right
which can be rewritten as
which gets all of the x's together in one place with only one thing happening to it, an exponential functon. To get rid of exponentials take logs of both sides. There are several ways to accomplish this. One way would be to change the exponential equation to logårithmic form.
Unfortunately you probably don't have a log4/3 button on your calculator. We will need to use the change of base formula
When we use the rule that the log of a quotient is the difference of the logarithms, we get
Which is the same answer we got before. Note that there is a one to one correspondence between the steps in the two methods. Each step in the arithmetic after you take logs is one level easier than the arithmetic before you take logs.
Punching this up on a calculator
log (4)/(log (4) - log (3)) = 4.818841679
Many scientific and graphing calculators will supply the left parentheses after the logs, but you will have to supply the right parentheses closing off these left parentheses. If you don't the calculator will probably thing that the rest of the expression is part of the argument in the log. You will, of course have to supply the parentheses around the bottom of the fraction. This is as many places as my TI-84 Plus will give me. Lgariithms are generally irrational numbers, so the decimals in this answer, theoretically, go on forever. Of course there is not enough time or space to list all of the infinitely many digits in the decimal expansion. One will need to round off. This calculator has rounded off to 10 significant digits.
T check with the decimal approximation, let us check x = 4.8188
34.8188 = 43.8188
Which the calculator gives as
34.8188 = 199.1369734 and = 43.8188 = 199.1345857
not quite the same thing. This illustrates that when one uses rounded off approximations, one cannot expect the result to be correct to more significant digits than the input. We rounded the input off to 5 significant digiits. Both of these answers really agree to 5 significant digits. If you use a more accurate approximation
34.8188417 = 43.8188417
34.8188417 = 199.1460965 whereas 43.8188417= 199.1460977
a better agreement. The more place of of accuracy in your rounded off approximation, the better the agreement in the check.
However, it is possible to check the answer without resorting to rounded off approximations. Copy down the original equation, except that instead of copying down the unknown, copy down what the unknown is equal to.
On the right we are adding fractions in the exponent. For that we will need common denominators.
Which still doesn't quite look right. The main problem is that we have different bases on the two sides of the equation. We can fix that by expressing both 3 and 4 as powers of 10.
When raising a power to a power, we multiply the powers. At that point, both of the exponents will have the same factors in their tops, and it checks.