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

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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.
  • #1
AzureNight
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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.
 
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  • #2
What have you tried so far? What is the definition of a logarithm?
 
  • #3
suggest you think about the inverse of logs of an integer base and how easily they can be worked out
 
  • #4
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.
 
  • #5
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
 
  • #6
o yea that should have been obvious, my bad
 

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

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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