Simply using laws of logarithms
2log8^2+2log8^3
I tried this question I failed
the coefficient in front of "log" has the same effect as the exponent in the argument
a log(b^c) = a c log(b)
2 log(8^2) = 2 * 2 * log(8) = 4 log(8)
2 log (8^3) = 2 * 3 * log(8) = 6 log(8)
2Log8^2 + 2Log8^3 = 2Log(8^2*8^3) = 2Log8^5.
To simplify the expression 2log8^2 + 2log8^3, we can use the basic laws of logarithms.
The first law we need to apply is the power rule, which states that log base a of b raised to the power c can be simplified as c times log base a of b. In other words, log base a of b^c = c * log base a of b.
Applying this rule to the given expression, we can rewrite it as:
2 * (log base 8 of 2^2) + 2 * (log base 8 of 3^3)
Next, let's simplify the expressions inside the parentheses.
Log base 8 of 2^2 simplifies to log base 8 of 4.
Log base 8 of 3^3 simplifies to log base 8 of 27.
Now, we have:
2 * log base 8 of 4 + 2 * log base 8 of 27
To further simplify, we can use another logarithmic property, which is the product rule. The product rule states that log base a of (bc) = log base a of b + log base a of c.
Applying the product rule to the expression, we get:
log base 8 of 4^2 + log base 8 of 27^2
This simplifies to:
log base 8 of 16 + log base 8 of 729
Now, we can evaluate the logarithms using common logarithms (base 10) or natural logarithms (base e).
log base 8 of 16 can be rewritten as log base 10 of 16 divided by log base 10 of 8.
Similarly, log base 8 of 729 becomes log base 10 of 729 divided by log base 10 of 8.
Evaluating the logarithms using a calculator, we get:
log base 10 of 16 ≈ 1.2041
log base 10 of 729 ≈ 2.463
Finally, substituting the values back into the expression, we have:
1.2041 + 2.463
Adding these values, we get approximately 3.6671.
Therefore, the simplified expression is approximately 3.6671.
Remember, when simplifying logarithmic expressions, it is crucial to apply the laws of logarithms correctly and work systematically to ensure accuracy.