Evaluate log3 108 + log3 100 - 2log3 20
Using the properties of logarithms, we can simplify the expression as follows:
log3 108 + log3 100 - 2log3 20
= log3 (108 * 100 / 20^2) (using the product and quotient rules of logarithms)
= log3 (108 * 100 / 400)
= log3 (27)
= 3 (since 3^3 = 27)
Therefore, log3 108 + log3 100 - 2log3 20 = 3.
Evaluate log4 (4 7/17) - log4 (3 21/68) + log4 48
3
I'm sorry, I need more information or context to understand what you are referring to. Can you please provide more details or clarify your question?
To evaluate the expression log3 108 + log3 100 - 2log3 20, we can use the logarithmic properties.
First, let's break down each term:
1. log3 108: This represents the logarithm of 108 to the base 3. To evaluate this, we need to find the power to which 3 must be raised to obtain 108. In other words, we need to solve the equation 3^x = 108. By solving this equation, we find that x = 4, since 3^4 = 81 and 3^5 = 243. So, log3 108 is equal to 4.
2. log3 100: This represents the logarithm of 100 to the base 3. Similarly, we need to find the power to which 3 must be raised to obtain 100. Solving the equation 3^x = 100, we find that x = 2, since 3^2 = 9 and 3^3 = 27. Therefore, log3 100 is equal to 2.
3. 2log3 20: This represents the logarithm of 20 to the base 3 multiplied by 2. Again, we need to find the power to which 3 must be raised to obtain 20. Solving the equation 3^x = 20, we find that x is approximately 2.726. Therefore, 2log3 20 is equal to approximately 2 × 2.726 ≈ 5.452.
Now, we can substitute these values back into the expression:
log3 108 + log3 100 - 2log3 20 = 4 + 2 - 5.452 ≈ 0.548
So, the value of the expression is approximately 0.548.
We can begin by rewriting the mixed numbers as improper fractions:
log4 (4 7/17) = log4 ((4*17+7)/17) = log4 (71/17)
log4 (3 21/68) = log4 ((3*68+21)/68) = log4 (237/68)
Substituting these values into the original expression, we get:
log4 (71/17) - log4 (237/68) + log4 48
Next, we can simplify the first two logarithms using the quotient rule of logarithms:
log4 (71/17 ÷ 237/68) + log4 48
Simplifying the fraction inside the first logarithm by multiplying by the reciprocal, we get:
log4 [(71/17) * (68/237)] + log4 48
Simplifying the expression inside the first logarithm using arithmetic, we get:
log4 (12) + log4 48
Using the product rule of logarithms, we can combine these two logarithms:
log4 (12 * 48)
Simplifying the product, we get:
log4 576
Finally, using the definition of logarithms, we see that:
log4 576 = x if and only if 4^x = 576
Multiplying 4 by itself repeatedly, we find that 4^3 = 64 and 4^4 = 256, so 4^3 * 4^3 = 4^6 = 4096. This means that:
4^4 < 576 < 4^5
Taking the logarithm base 4 of both sides of this inequality, we get:
4 < log4 576 < 5
Therefore, log4 576 is between 4 and 5, so it must be approximately 4.292. Therefore:
log4 (4 7/17) - log4 (3 21/68) + log4 48 ≈ 4.292.