Question

Write the expression as a single logarithm

Answer 1

\[3\log_{2}y-\frac{ 1 }{ 2 } \log_{2}w+4\log_{2}x \]

Answer 2

the first step is "bring in" the coefficient out front, using this rule
\[ a \log b = \log b^a \]

Answer 3

Okay so like this?
\[\log_{2}y^3 \]

Answer 4

yes. do that for the other terms also.

Answer 5

\[\log_{2}y^3 - \log_{2}w^\frac{ 1 }{ 2 } + \log_{2}x^4 \]

Answer 6

notice all the logs have the same base.
that means if you add two of them, you can "merge" them
but multiply instead of add inside the log

in other words, use this rule
\[ \log a + \log b = \log(a\cdot b) \]

Answer 7

if you subtract, you divide inside:
\[ \log a - \log b = \log \frac{a}{b} \]

Answer 8

Okay\[\log_{2}y^3-\log_{2}w^\frac{ 1 }{ 2 } = \log_{2}\frac{ y^3 }{ w^\frac{ 1 }{ 2 } } \]

Answer 9

Is that right? (I just did the first part, for now)

Answer 10

yes

Answer 11

Okay\[\log_{2}\frac{ y^3 }{ w^\frac{ 1 }{ 2 } }+\log_{2}x^4=\log_{2}(\frac{ y^3 }{ w^\frac{ 1 }{ 2 } })(x^4) \]

Answer 12

Correct?

Answer 13

the x^4 is "inside" so you should write it as
\[ \log_2\left(\frac{y^3x^4}{w^\frac{1}{2}} \right)\]
or, if this is multiply choice, they might try to confuse you and write the 1/2 power as a square root:
\[ \log_2\left(\frac{y^3x^4}{\sqrt{w}} \right)\]

Answer 14

Oh, okay. (not a multiple choice,btw)

Answer 15

Is that the all, or is there more to it?

Answer 16

that is all you have to do. It is now one logarithm of a messy expression

Answer 17

Okay thank you! I thought so, but I just wanted to be sure!