1/2 log 8 + log 32 - log 2 all in base 4

Since 2 = 4^(1/2)

8 = 2^3 = 4^(3/2)
32 = 2^5 = 4^(5/2)

So, you have 1/2 * 3/2 + 5/2 - 1/2 = ____

Thank you

This is an eye opener i never knew it could be solved this way

Why did the math book go to the comedy club? To solve logarithmic equations with a little bit of humor, ba dum tss!

Now, let's tackle your equation. To solve this expression, we can use the properties of logarithms. First, let's simplify it step by step:

1/2 log₄(8) + log₄(32) - log₄(2)

First, let's simplify the individual logarithmic terms:

log₄(8) = 3, since 4^3 = 8
log₄(32) = 5, since 4^5 = 32
log₄(2) = 1/2, since 4^(1/2) = 2

Now we can substitute these values back into the original equation:

1/2(3) + 5 - 1/2

This simplifies to:

3/2 + 5 - 1/2

Combine like terms:

(3/2 - 1/2) + 5

2/2 + 5

1 + 5

So, the final answer is:

6

Hope that adds up and brightens your day!

To simplify the expression 1/2 log 8 + log 32 - log 2 in base 4, we can utilize the logarithmic properties.

First, let's convert the given expression into a single logarithm using the rules of logarithms:

1/2 log 8 + log 32 - log 2

Using the rule log a + log b = log(ab), we can rewrite this as:

log √8 + log 32 - log 2

Next, let's apply the rule log a - log b = log(a/b) to combine the logarithms further:

log (√8 × 32 / 2)

Simplifying the expression inside the logarithm:

log (√8 × 32 / 2)
= log (√(8 × 32) / 2)
= log (√256 / 2)
= log (16 / 2)
= log 8

Finally, we have a single logarithm, log 8. Since we want the expression in base 4, we can rewrite it as:

log 8 = log 8 / log 4

Using the change of base formula:

log 8 / log 4 = log₄ 8

So, the simplified expression in base 4 is log₄ 8.