SOLUTION: Evaluate each expression without using a caculator: Log_2X+Log

SOLUTION: Evaluate each expression without using a caculator: Log_2X+Log_4X=6

Question 277175: Evaluate each expression without using a caculator:
Log_2X+Log_4X=6
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
What do the underscores, "_", mean?

And is the equation

or
?

Please repost your question making it clear. If needed use more English and less algebraic notation. For example the second equation would be:
base 2 log of x + base 4 log of x = 6
*******
Now that I know what the equation is...

To solve equations where the variable is in the argument(s) of logarithms, you usually start by transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)

Your equation, with the "non-logarithmic" term of 6 on the right, makes the second form more difficult to achieve. So we will aim for the first form.

The first form requires that one side is a single logarithm. So somehow we need to combine your two logarithms into one. Your two logarithms are not like terms so they cannot be added. In addition, a property of logarithms, , can be used to combine two logarithms which have a "+" between them. But this property requires that the bases of the two logarithms be the same. And your bases are different so we cannot use this property (yet).

So we need the bases the same. Fortunately there is a base conversion for logarithms, , which can be used to convert a logarithm of one base, "a", into an expression of another base, "b". We will use this to convert your base 4 logarithm into base 2:

And the denominator is a logarithm we can do "by hand". Since , :

These two logarithms are like terms so we can go ahead and add them:

The only thing left to do in order to achieve the desired form is the get rid of the 3/2. We can accomplish this by multiplying both sides by 2/3:

We finally have the desired form. With this form the next step to rewrite the equation in exponential form:

which simplifies to

And we have the answer.

Checking the answer is important (not just a good idea) with logarithmic equations. Always check using the original equation:

Checking x = 16:

Since and , and . This gives us:
Check.
RELATED QUESTIONS
Evaluate each expression without using a calculator. a) log 1000 b) log 1 c) log2 +... (answered by josgarithmetic,MathTherapy)

evaluate without using a calculator:... (answered by stanbon)

evaluate √log(100)-log(√100) without using a... (answered by lwsshak3)

Can you please help to solve the following: Assume that log 4 ≈ 0.0602, log 5... (answered by stanbon,MathLover1)

SOMEONE HELP PLEASE: Evaluate or simplify the expression without using a calculator.... (answered by KnightOwlTutor)

Evaluate the expression without using a calculator 4 log(3)... (answered by stanbon)

evaluate the expression without using a calculator. [log]_3... (answered by lynnlo)

Evaluate the following expression without using a calculator. {{{8^log(8,(19))}}} (answered by Edwin McCravy,ikleyn)

Can you please help to solve the following: Assume that log 4 ≈ 0.0602, log 5... (answered by stanbon,richard1234)