Question

(〖log〗_4 10)/(〖log〗_4 9)+〖log〗_9 0,1

Answer 1

㏒₄10

----------- = ㏒₉10

㏒₄9

㏒₉10+㏒₉0,1= ㏒₉10*0,1 =㏒₉1 =0

Answer 2

Let's break down the expression step by step:

1. (log₄10) / (log₄9): First, let's simplify the expression inside the parentheses. Recall that logₐb represents the logarithm of b with base a. So,

log₄10: This represents the logarithm of 10 with base 4. log₄9: This represents the logarithm of 9 with base 4.

To calculate these logarithms, we need to find out what power we have to raise 4 to, in order to get 10 and 9, respectively.

4^1 = 4 4^2 = 16 4^3 = 64

From this, we can see that 10 lies between 4^1 and 4^2 (between 4 and 16), so log₄10 will be between 1 and 2. Similarly, 9 is equal to 4^2, so log₄9 is equal to 2.

Now, we can calculate the expression (log₄10) / (log₄9):

(log₄10) / (log₄9) ≈ 1 / 2 = 0.5

2. log₉(0.1): Here, we have the logarithm of 0.1 with base 9. We need to find out what power we have to raise 9 to, in order to get 0.1.

9^1 = 9 9^2 = 81 9^3 = 729

0.1 lies between 9^1 and 9^2 (between 9 and 81). Since 0.1 is less than 1, the logarithm will be negative. Specifically, log₉(0.1) will be between -1 and -2.

Now, we can calculate the expression log₉(0.1):

log₉(0.1) ≈ -1.x (where x is some decimal value)

So, the final expression is:

0.5 + log₉(0.1) ≈ 0.5 - 1.x ≈ -0.5 - x

Without knowing the exact values of the logarithms, we can't provide an exact numerical result, but it will be approximately equal to -0.5 (since the x value is small).