# Solving Logarithmic Equations: Find log[base 2]log[base 3]a = 2

• AzureNight
In summary, the conversation is about finding the value of a in the equation log[base 2](log[base 3]a) = 2. The participants discuss the definition of a logarithm and suggest using the inverse of logs of an integer base to solve the equation. Finally, it is determined that a = 81.
AzureNight
I have a question I need to do which I'm stuck on:

find a

log[base 2](log[base 3]a) = 2

How would I do that? And sorry about the crude representation of the base, but I'm not sure how to do a small 2 or 3 on the computer.

Edit: oh damn, wrong forum. I guess it has to be moved to the appropriate one for homework questions.

Last edited:
What have you tried so far? What is the definition of a logarithm?

suggest you think about the inverse of logs of an integer base and how easily they can be worked out

Ok well. log[base a] x = y then a^y= x, by definition.

Then in this case, 2^2 = log[base 3] a
log[base3]a = 4

Indentity- log[base x] y = (ln y)/(ln x) Hopefully uve learned this.

ln a = 4 ln 3

make both sides the exponents of e.
a= e^(4 ln3 )

Therefore, a = 81.

Gib Z said:
Ok well. log[base a] x = y then a^y= x, by definition.

Then in this case, 2^2 = log[base 3] a
log[base3]a = 4

Indentity- log[base x] y = (ln y)/(ln x) Hopefully uve learned this.

ln a = 4 ln 3

make both sides the exponents of e.
a= e^(4 ln3 )

Therefore, a = 81.
no need for e or your identity, just use your 1st line defn again

o yea that should have been obvious, my bad

## 1. How do you solve logarithmic equations?

To solve logarithmic equations, you can use the basic property of logarithms which states that log[base b]x = y if and only if b^y = x. This means that to solve an equation such as log[base 2]x = 3, you can rewrite it as 2^3 = x, giving you the answer of x = 8.

## 2. What is the meaning of "log" in logarithmic equations?

Log in logarithmic equations stands for logarithm, which is a mathematical function that calculates the power to which a base number must be raised to produce a given number. For example, log[base 2]8 = 3 means that 2^3 = 8.

## 3. What is the base of a logarithm?

The base of a logarithm is the number that is raised to a certain power to produce a given number. In the equation log[base 2]8 = 3, the base is 2.

## 4. How do you solve equations with multiple logarithms?

To solve equations with multiple logarithms, you can use the basic properties of logarithms such as log[base b](x*y) = log[base b]x + log[base b]y and log[base b](x/y) = log[base b]x - log[base b]y. You can also use the change of base formula, log[base b]x = log[base c]x / log[base c]b, to simplify the equation and solve for the variable.

## 5. What is the relationship between logarithms and exponents?

Logarithms and exponents are inverse functions of each other. This means that if the logarithm of a number is y, then the exponent of that number is x. For example, log[base 2]8 = 3 means that 2^3 = 8. This relationship allows us to solve logarithmic equations by using exponents.

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