Question

Log3 135- log9 400 + 2log3 6

Answer 1

$log_3135-log_9400+2log_36 = log_3135-\frac{1}{2}log_3400+log_336 = \\ \\ = log_3135-log_320+log_336 = log_3\frac{135*36}{20} = log_3243 = log_33^5 = 5$

Answer 2

To simplify the expression, let's use the following logarithmic properties:

1. log_a(b) - log_a(c) = log_a(b/c)
2. n * log_a(b) = log_a(b^n)

Now, let's simplify the given expression step by step:

1. log3(135) - log9(400) + 2 * log3(6)

Step 1: Apply the first property to combine the logarithms with the same base: log3(135) - log3(400^(2/2)) + 2 * log3(6)

Step 2: Simplify the logarithms with the same base: log3(135) - log3(20) + 2 * log3(6)

Step 3: Apply the second property to remove the coefficients: log3(135) - log3(20) + log3(6^2)

Step 4: Simplify further: log3(135) - log3(20) + log3(36)

Now, let's evaluate the logarithms:

Step 5: Use the definition of logarithms to find the values: log3(135) ≈ 3.815 log3(20) ≈ 2.726 log3(36) ≈ 3.170

Step 6: Substitute the values back into the expression: 3.815 - 2.726 + 3.170

Step 7: Finally, perform the arithmetic: ≈ 4.259

So, the simplified value of the expression log3(135) - log9(400) + 2 * log3(6) is approximately 4.259.