Solving Logarithms: Discovering the Unknown Variable in Log_a (100)

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In summary, the conversation discusses the problem of solving for log_a(100) with the given conditions of log_a(2) = 20 and log_a(5) = 30. However, the solution is inconsistent as it leads to 5^2 = 2^3, which is false.
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How do i solve Log_a (100) if Log_a (2) = 20 and Log_a (5) = 30

I got to 2^(1/20) = 100^(1/x) and 5^(1/30) = 100^(1/x) but didnt know how to go any further.
 
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Luis Melo said:
How do i solve Log_a (100) if Log_a (2) = 20 and Log_a (5) = 30

I got to 2^(1/20) = 100^(1/x) and 5^(1/30) = 100^(1/x) but didnt know how to go any further.

If log_a(x) means log to base "a" of x, then the two conditions you gave are inconsistent: you get two different values of "a" in the two cases.
 
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Luis Melo said:
How do i solve Log_a (100) if Log_a (2) = 20 and Log_a (5) = 30

I got to 2^(1/20) = 100^(1/x) and 5^(1/30) = 100^(1/x) but didnt know how to go any further.

Is your question "What is [itex]\log_a(100)[/itex] if [itex]\log_a(2) = 20[/itex] and [itex]\log_a(5) = 30[/itex]"?

Well, you can get the answer from the fundamental principle of logarithms: [itex]\log_a(xy) = \log_a(x) + \log_a(y)[/itex].

However this is a spectacularly poorly designed question, since it asserts that [itex]a^{20} = 2[/itex] and [itex]a^{30} = 5[/itex], which together require [itex]5^2 = 2^3[/itex]. This is plainly false.
 
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Thank you or the answers.
 

FAQ: Solving Logarithms: Discovering the Unknown Variable in Log_a (100)

1. What is a logarithm?

A logarithm is the inverse operation of exponentiation. It is used to solve exponential equations by finding the unknown variable.

2. How do you solve a logarithm?

To solve a logarithm, you must isolate the logarithmic expression on one side of the equation and rewrite it in exponential form. Then, you can solve for the unknown variable by using the properties of exponents and taking the logarithm of both sides.

3. What is the base of a logarithm?

The base of a logarithm is the number that is raised to a certain power to equal the argument of the logarithm. In the equation log_a (x), a is the base.

4. Why do we need to use logarithms?

Logarithms are useful for solving exponential equations, which often arise in mathematical and scientific problems. They also help to condense large numbers into more manageable values.

5. How do you solve a logarithm with a variable in the base?

If the variable is in the base of the logarithm, you can use the change of base formula to rewrite it in terms of a different base. Then, you can solve for the unknown variable using the techniques mentioned before.

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