2log3+4log2-3
To solve this expression, we need to use the properties of logarithms.
First, we can apply the power rule of logarithms, which states that log_b(x^a) = alog_b(x). Using this rule, we can rewrite the expression as:
2log3 + 4log2 - 3 = log3^2 + log2^4 - log3^3
Next, we can apply the product rule of logarithms, which states that log_b(xy) = log_b(x) + log_b(y). Using this rule, we can simplify the expression further:
log3^2 + log2^4 - log3^3 = log3^2 + log2^4 - log(3^3)
Now, we can use the power rule of logarithms again to simplify:
log3^2 + log2^4 - log(3^3) = log9 + log16 - log27
Finally, we can use the addition/subtraction rule of logarithms, which states that log_b(x) ± log_b(y) = log_b(x*y). Using this rule, we can simplify the expression one last time:
log9 + log16 - log27 = log(9 * 16) - log27
Now, we can calculate the final value:
log(9 * 16) - log27 = log(144) - log27
Using logarithmic identities, we can simplify further:
log(144) - log27 = log(144/27)
Now, we can simplify the division:
log(144/27) = log(144) - log(27)
Using a calculator to calculate the logarithms of 144 and 27, we get:
log(144) ≈ 2.158
log(27) ≈ 1.431
Therefore, the final value of the expression is approximately:
2.158 - 1.431 ≈ 0.727