Not understanding Logaritims

Question

Given that log a ( 4 ) ≈ 0.58 and log a ( 5 ) ≈ 0.67 , evaluate each of the following. Hint: use the properties of logarithms to rewrite the given logarithm in terms of the the logarithms of 4 and 5.

a)  loga(0.8)≈

b)  loga(3.2) ≈

c)  loga(64)≈

d)  loga(√5)≈

e)  loga(100)≈

f)  loga(20)≈

Answer 1

I'm sorry this is a late reply. Although the other response here isn't wrong in what it says about logarithms, I don't believe it's really addressing the homework the way it was intended. The hint is very clear that you should be using the properties of logarithms to rewrite the ones you're asked to solve in terms of the log facts you were given about the logarithms for 4 and 5.

Okay, so what are the properties of logarithms they care about?

log_a (uv) = log_a u + log_a v

log_a (u / v) = log_a u - log_a v

log_a uⁿ = n log_a u

Let's see if we can use them to answer some of these. I'm going to start with

f. log_a (20)≈

log_a (20) = log_a (4*5)

= log_a (4) + log_a (5) (from the first logarithm property above)

≈ .58 + .67

≈ 1.25

So the trick here is finding ways to express each of the questions you're asked in terms of multiplication, division, or exponentiation of 4 and 5. I won't work out the answers for you, but give you some strong hints:

e)  log_a (100)≈ log_a (4*5*5)

d)  log_a (√5) = log_a (5^1/2) ≈ 1/2 * log_a (5)

c) log_a (64) = log_a (4³) (you could also think of this as log_a (64) = log_a (4*4*4))

b) log_a (3.2) = log_a (32/10) = log_a (16/5) = log_a ((4*4)/5)

a) log_a (.8) = log_a (8/10) = log_a (4/5)

Answer 2

A logarithm is basically just an exponent. Loga(5) = 0.67 simply means that a^0.67 = 5 or a^2/3 = 5. This can be simplified to (3√a)² = 5 so if you work that out, then a is approximately 11.18. You can check it by substituting the log from the first equation - 11.18^0.58 = 4. For the remaining equations, you can use 11.18 to solve, but for d, you can use a log property (logaUⁿ = nlogaU) and rewrite that as 1/2loga(5), since an exponent of 1/2 is the same as a square root.