# I can't remember how to solve equations with logarithms/exponents

• Jamin2112
In summary, the conversation revolved around solving a logarithm problem involving the equation t2/t1 = (d2/d1)1.06 and finding a solution for d2. Different approaches were suggested, including using logarithmic identities and changing the subject of the formula, but the most efficient approach involved taking the natural log of both sides and using exponentiation to isolate d2.
Jamin2112
This is frustrating me so much. I've been out of college for 1 year and I already forgot how to solve logarithm problems (though, to my surprise, I've encountered one I need to solve in real life.)

t2/t1 = (d2/d1)1.06

and I need to solve for d2.

I know the equation is the equivalent of

logd2/d1(t2/t1) = 1.06,

but I still can't figure out how to isolate the d2.

Jamin2112 said:
This is frustrating me so much. I've been out of college for 1 year and I already forgot how to solve logarithm problems (though, to my surprise, I've encountered one I need to solve in real life.)

t2/t1 = (d2/d1)1.06

and I need to solve for d2.

I know the equation is the equivalent of

logd2/d1(t2/t1) = 1.06,

but I still can't figure out how to isolate the d2.
Start by taking the log of both sides.

Mark44 said:
Start by taking the log of both sides.

log1.06 of both sides?

log10 or ln, either one.

Raise both sides to the 1/1.06 power.

Chet

Chestermiller said:
Raise both sides to the 1/1.06 power.

Wow, I feel like an idiot now. Was Mark44 trying to lead me down a rabbit hole?

Jamin2112 said:
Wow, I feel like an idiot now. Was Mark44 trying to lead me down a rabbit hole?
No. Mark is a serious guy. He had a different approach in mind, probably motivated by your questions about logarithms.

Chet

Chestermiller said:
No. Mark is a serious guy. He had a different approach in mind, probably motivated by your questions about logarithms.

When I originally tried doing the "log to both sides, ..." approach, I started going in circles.

Starting with logs seems like the hard way to me. Do what you usually do when you want to change the subject of a formula: get d2 on its own on one side.

First use the exponent rule: (a/b)^n = a^n/b^n, then re-arrange to get

d_2^1.06 = ...

Then continue.

EDIT: 'log rule' changed to 'exponent rule'

Last edited:
qspeechc said:
Starting with logs seems like the hard way to me. Do what you usually do when you want to change the subject of a formula: get d2 on its own on one side.

First use the log rule: (a/b)^n = a^n/b^n, then re-arrange to get
That would be an exponent rule.
qspeechc said:
d_2^1.06 = ...

Then continue.
I agree that this approach is simpler, but taking logs of both sides isn't that much longer. After you take the natural log of both sides of the original equation, you have
$$ln(\frac{t_2}{t_1}) = 1.06 ln(\frac{d_2}{d_1})$$
Now divide both sides by 1.06 and exponentiate to get d2/d1 by itself. One more step and you're done.

The approach I suggested was just the first one to come to mind.

Slip of the tongue, so to speak ...:p

## 1. How do I solve equations with logarithms?

To solve equations with logarithms, you first need to isolate the logarithm on one side of the equation. Then, use the inverse operation of logarithms, which is exponentiation, to eliminate the logarithm and solve for the variable.

## 2. What is the difference between logarithms and exponents?

Logarithms and exponents are inverse operations of each other. While exponents represent repeated multiplication, logarithms represent repeated division. For example, in the expression 23, 3 is the exponent and 8 is the result. In the expression log28, log is the logarithm and 3 is the result.

## 3. How do I know when to use logarithms in an equation?

Logarithms are typically used when the variable is in an exponent or when the equation involves exponential growth or decay. You can also use logarithms to solve for an unknown exponent in an equation.

## 4. What are the properties of logarithms that I need to know?

The three main properties of logarithms are the product rule, quotient rule, and power rule. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual terms. The power rule states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number.

## 5. How can I check my solution to an equation with logarithms?

You can check your solution by plugging it back into the original equation and simplifying both sides. If the resulting values are equal, then your solution is correct. You can also use a calculator to graph both sides of the equation and see if they intersect at the same point, indicating a correct solution.

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