Without using technology, rank these logarithms in order from least to greatest: log₄62,log₆36,log₃10,log₅20. I'm not sure how to do this question because the bases are different. Thanks!
log₄62 = a , 4^a = 62 , I know 4^3 = 64 , so a = appr 3 , but a < 3
log₆36 = b , 6^b = 36, so b = 2
log₃10 = c , 3^c = 10, I know 3^2 = 10 , so c = appr 2, but c >2
log₅20 = d , 5^d = 20 , I know 4^2 = 25, so d < 2
order: log₅20 , log₆36 , log₃10 , log₄62
To rank logarithms with different bases without using technology, we can use the change of base formula. The change of base formula states that for any positive numbers a and b, and any positive number x:
logₐx = log_bx / log_ba
We can use this formula to rewrite the logarithms in a common base (let's use base 10). I will solve each logarithm step-by-step:
1. log₄62:
Since we're using the change of base formula, we can rewrite this as:
log₄62 = log₁₀62 / log₁₀4
Now, let's compute each of the logarithms:
log₁₀62 ≈ 1.792
log₁₀4 ≈ 0.602
Therefore, log₄62 ≈ 1.792 / 0.602 ≈ 2.982
2. log₆36:
Again, using the change of base formula, we have:
log₆36 = log₁₀36 / log₁₀6
Computing each of the logarithms:
log₁₀36 = 1.556
log₁₀6 ≈ 0.778
Therefore, log₆36 ≈ 1.556 / 0.778 ≈ 2.000
3. log₃10:
Applying the change of base formula:
log₃10 = log₁₀10 / log₁₀3
Computing each of the logarithms:
log₁₀10 = 1.000
log₁₀3 ≈ 0.477
Therefore, log₃10 ≈ 1.000 / 0.477 ≈ 2.094
4. log₅20:
Using the change of base formula:
log₅20 = log₁₀20 / log₁₀5
Computing each of the logarithms:
log₁₀20 ≈ 1.301
log₁₀5 ≈ 0.699
Therefore, log₅20 ≈ 1.301 / 0.699 ≈ 1.860
Now, we can rank the logarithms from least to greatest:
log₅20 < log₆36 < log₄62 < log₃10
To rank logarithms with different bases from least to greatest without using technology, we need to use some logarithmic properties and convert them to a common base. Here's how to do it step by step:
Step 1: Express all the logarithms with different bases in terms of a common base. In this case, let's choose base 10.
- Use the change of base formula to convert each logarithm to base 10:
- log₄62 = log₁₀62 / log₁₀4
- log₆36 = log₁₀36 / log₁₀6
- log₃10 = log₁₀10 / log₁₀3
- log₅20 = log₁₀20 / log₁₀5
Step 2: Find the decimal approximations of each logarithm using the common base.
- Calculate each numerator and denominator separately using either mental math or basic calculations:
- log₁₀62 = 1.792 (approximately)
- log₁₀4 = 0.602 (approximately)
- log₁₀36 = 1.556 (approximately)
- log₁₀6 = 0.778 (approximately)
- log₁₀10 = 1 (exact)
- log₁₀3 = 0.477 (approximately)
- log₁₀20 = 1.301 (approximately)
- log₁₀5 = 0.699 (approximately)
Step 3: Substitute the decimal approximations back into the original logarithms.
- log₄62 ≈ 1.792 / 0.602
- log₆36 ≈ 1.556 / 0.778
- log₃10 ≈ 1 / 0.477
- log₅20 ≈ 1.301 / 0.699
Step 4: Simplify each expression.
- log₄62 ≈ 2.979
- log₆36 ≈ 2.000
- log₃10 ≈ 2.093
- log₅20 ≈ 1.860
Step 5: Rank the logarithms from least to greatest based on their decimal approximations.
1. log₅20 ≈ 1.860
2. log₆36 ≈ 2.000
3. log₃10 ≈ 2.093
4. log₄62 ≈ 2.979
So, the logarithms from least to greatest are: log₅20, log₆36, log₃10, log₄62.