# Logarithmic Question: Solving for Unknowns Using Logarithm Properties"

• gl0ck
In summary, the student is struggling to solve a homework equation involving logs and is missing a right parenthesis. They try to solve it using properties of logs, but end up with an incorrect answer.
gl0ck

## Homework Statement

Hi, it maybe stupid question, but I struggle on this problem..
log16(4354)+3/log25(43)=a+b*log25

## Homework Equations

Using properties:
logabn=n logab
logn(bc) = lognb+lognc
logba=1/logab

## The Attempt at a Solution

I can't find the b number.
log16(4354)+3/log25(43)=6 log162+4log165+3/3log254=6(log22/log216)+4(log25/log216)+1/log254=3/2 + 2log252= 3/2 + 4log25

so a=3/2 b=4 but in the answer they find b to be 2 not 4
Can you please tell me where is my mistake?

Thanks

Last edited:
gl0ck said:

## Homework Statement

Hi, it maybe stupid question, but I struggle on this problem..
log16(4354+3/log25(43)=a+b*log25
You're missing a right parenthesis, so it's hard to tell what you're starting with.

Did you mean log16(4354) +3/log25(43)?
gl0ck said:

## Homework Equations

Using properties:
logabn=n logab
logn(bc) = lognb+lognc
logba=1/logab

## The Attempt at a Solution

I can't find the b number.
log16(4354+3/log25(43)=6 log162+4log165+3/3log254=6(log22/log216)+4(log25/log216)+1/log254=3/2 + 2log252= 3/2 + 4log25

so a=3/2 b=4 but in the answer they find b to be 2 not 4
Can you please tell me where is my mistake?

Thanks

Lol, yes, Sorry it is definitely log16(4354)+3/log25(43)

Can we at least presume you know the basic properties of logarithms and so know that $log_{16}(4^35^4)= 3log_{16}(4)+ 4 log_{16}(5)$? And, of course, since $16= 4^2$, $4= 16^{1/2}$ so that $log_{16}(4)= 1/2$. That is, $$log_{16}(4^35^4)= 3+ 4log_{16}(5)$$
$log_{25}(4^3)= 3log_{25}(4)$.

Perhaps what you are missing is that $\log_a(x)= \frac{log_b(x)}{log_a(b)}$. You can use that to change the $log_{25}$ to $log_{16}$. Further, since, $16= 2^4$, $log_{2}(16)= 4$ and so $log_{16}(x)= \frac{log_2(x)}{4}$

You can use that to change everything to $log_2$ as you have on the right.

Last edited by a moderator:
HallsofIvy said:
Perhaps what you are missing is that $\log_a(x)= \frac{log_b(x)}{log_a(b)}$

I'm noob, but I think that you meant
$\log_a(x)= \frac{log_b(x)}{log_b(a)}$
Mainly because if you start with one base and end with two it was not a change of base.
I may be wrong.

besulzbach said:
I'm noob, but I think that you meant
$\log_a(x)= \frac{log_b(x)}{log_b(a)}$
Mainly because if you start with one base and end with two it was not a change of base.
I may be wrong.

Don't worry, you're correct

1 person

## 1. What is a logarithm?

A logarithm is a mathematical function that helps us solve equations in which the unknown value is an exponent. It is written as logb(x), where b is the base and x is the result of the exponent.

## 2. What are the properties of logarithms?

There are three main properties of logarithms: the product rule, the quotient rule, and the power rule. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual numbers. 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.

## 3. How do I solve an equation using logarithm properties?

First, identify which property of logarithms can be applied to the equation. Then, use the property to rewrite the equation in a simpler form. Continue simplifying the equation until the unknown value is isolated. Finally, solve for the unknown value using basic algebraic principles.

## 4. Can logarithm properties be used to solve real-world problems?

Yes, logarithm properties can be used to solve real-world problems involving exponential growth or decay, such as population growth, compound interest, or radioactive decay. They can also be used to solve equations in physics, chemistry, and other scientific fields.

## 5. Are there any restrictions when using logarithm properties?

Yes, there are a few restrictions to keep in mind when using logarithm properties. The base of the logarithm must be positive and not equal to 1. The argument (the number inside the parentheses) of a logarithm must be positive. And when using the quotient rule, the denominator cannot be 0.

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