Expressing a log function with a different argument/base.

[Banks]

Hi there,

I'm struggling with logs, so would appreciate some help on the below questions:

1. If log5(3) = T, express log5(9/125) in terms of T

2. If log7(12) = x, express log10(12) in terms of x.

Could someone please kindly advise on how I'd do this?

Thank you in advance

 

[mmm4444bot]

I'm struggling with logs, so would appreciate some help on the below questions:

1. If log5(3) = T, express log5(9/125) in terms of T

2. If log7(12) = x, express log10(12) in terms of x.

For exercise (1), can you express log₅(9/125) as a difference? There's a property for that.

You can also make use of the property: log(a^n) = n*log(a)

For exercise (2), start by using the Change-of-Base Formula, to express log₇(12) in terms of log-base10. See if you can continue.

 

[Banks]

For exercise (1), can you express log₅(9/125) as a difference? There's a property for that.

You can also make use of the property: log(a^n) = n*log(a)

For exercise (2), start by using the Change-of-Base Formula, to express log₇(12) in terms of log-base10. See if you can continue.

Thanks for replying.

Apologies, I'm struggling with logs a little!

Q1: log5(3) - log5(9/125) = log5(3 / (9/125) - is this correct? from here, how would I express in terms of T? is it just the answer to: log5(3 / (9/125) = T?

Q2: log7(12) = log10(12) / log10(7) - is this right? so, the answer would be: 1.276989 = x (since it should be in terms of x)?

Thank you very much.

 

[tkhunny]

Q2: You have told us what \(\displaystyle \log_{7}(12)\) is, but you have not done what was asked.

You have this:
\(\displaystyle \log_{7}(12) = \dfrac{\log_{10}(12)}{\log_{10}(7)} = x\)

Now, reread the problem statement. What are you asked to do?

 

[JeffM]

Thanks for replying.

Apologies, I'm struggling with logs a little!

Q1: log5(3) - log5(9/125) = log5(3 / (9/125) - is this correct? from here, how would I express in terms of T? is it just the answer to: log5(3 / (9/125) = T?

\(\displaystyle \text {Given: } t = log_5(3) \implies log_5 \left ( \dfrac{9}{125} \right ) = log_5(9) - log_5(125). = WHAT?\)

It is a general property of logarithms that

\(\displaystyle log_a \left ( \dfrac{b}{c} \right ) = log_a(b) - log_a(c).\)

You need to memorize the properties of logs and use them.

To finish solving this problem requires still yet another property of logs.

 

[mmm4444bot]

Apologies, I'm struggling with logs a little!

We've all been there; no need to apologize.

log5(3) - log5(9/125) = log5(3 / (9/125) -- is this correct?

No -- that would be finding the difference of the two given logs; we're asked to do something else. Let's look again at the instruction given:

express log₅(9/125) in terms of T

We need to find a way to rewrite log₅(9/125) as an expression containing a log₅(3) term somewhere (to be replaced with symbol T, later).

9/125 is a ratio, so we could try using the property that deals with the log of a ratio:

log(a/b) = log(a) - log(b)

We get log₅(9) - log₅(125)

We don't see any log₅(3) terms, yet. Now continue.

Think about the term log₅(9). What kinds of things can we do with it?

Think about the term log₅(125). What does it mean?

log7(12) = log10(12) / log10(7) -- is this right? so, the answer would be: 1.276989 = x (since it should be in terms of x)?

When we are asked to express some quantity "in terms of x", it means we need to find a symbolic expression that contains symbol x. We don't need to find a numerical value for x.

Your first step is good.

log₇(12) = x

log₁₀(12) / log₁₀(7) = x

We're asked to express log₁₀(12) as something, so solve that last equation for log₁₀(12). In other words, get log₁₀(12) all by itself, on one side of the equation, so that you can see what it equals.

 

[stapel]

I'm struggling with logs, so would appreciate some help on the below questions:

1. If log5(3) = T, express log5(9/125) in terms of T

2. If log7(12) = x, express log10(12) in terms of x.

Could someone please kindly advise on how I'd do this?

It might help to review log rules (here), as well as looking at some common "trick" questions which require that you understand how those rules work (here).

For question (1), what does the change-of-base formula give you for "\(\displaystyle \log_5\left(\frac{9}{125}\right)\, =\, \log_5\left(\frac{3^2}{5^3}\right)\)"? Where does this lead?

For question (2), use the change-of-base formula, and solve for the specified portion.