Question

Известно что log2 5=a log 2 3=b найдите log16 45

Answer 1

$log_{16} 45=(log_{2} 45)/(log_{2} 16)=(log_{2} (5*9))/(log_{2} 2^4)=\\ \\ (log_{2} 5+log_{2} 3^2)/4=\frac{a+2b}{4}$

Answer 2

To find log₁₆ 45, we can use the properties of logarithms to express it in terms of the given values of a and b.

The property we'll use is:

logᵦ (x) = logᵦ (a) * logₐ (x)

We know that log₂ 5 = a and log₂ 3 = b. Now, we need to find log₁₆ 45.

Step 1: Express 45 in terms of 16, 3, and 5 Since 45 = 3 * 3 * 5, we can rewrite 45 as:

45 = 3² * 5

Step 2: Apply the property of logarithms Now, using the property mentioned above, we have:

log₁₆ 45 = log₁₆ (3² * 5)

Step 3: Split the logarithm using the properties of logarithms Using another property of logarithms, we can split the logarithm:

log₁₆ (3² * 5) = log₁₆ (3²) + log₁₆ 5

Step 4: Express the terms using the given information Now, we already know that log₂ 3 = b, so we can convert the first term:

log₁₆ (3²) = log₂ (3²) / log₂ (16) = 2 * b / 4 = b / 2

Step 5: Simplify and find the value of log₁₆ 45 Putting it all together:

log₁₆ 45 = b / 2 + log₁₆ 5

We are still missing the value of log₁₆ 5. However, we can express it using the given information:

log₁₆ 5 = log₂ 5 / log₂ 16 = a / 4

Finally, we have:

log₁₆ 45 = b / 2 + a / 4

So, to find the numerical value of log₁₆ 45, you need the actual values of a and b, which are not provided in the question. Once you have the values of a and b, you can calculate log₁₆ 45 using the equation above.