3log2+log20 _ log 1.6
log 2^3 + log20 - log1.6
log(8*20/1.6) = log100 = 2
or
3log2 + log(2^2 * 5) - log(2^4/10)
3 log2 + (2log2 + log5) - (4log2 - log10)
log2+log5 + log10
2log10
2
To simplify the expression 3log2 + log20 - log1.6, we can use several logarithmic properties:
1. Power Rule: log_a(b^c) = c * log_a(b)
2. Product Rule: log_a(b * c) = log_a(b) + log_a(c)
3. Quotient Rule: log_a(b / c) = log_a(b) - log_a(c)
4. Logarithmic Identity: log_a(a) = 1
Let's simplify each term step by step:
1. 3log2 = log2(2^3) (using the power rule)
= log2(8) (simplifying the exponent)
= log2(2^3) = 3 (using the logarithmic identity)
2. log20 = log2(2 * 10) (using the product rule)
= log2(2) + log2(10) (simplifying the expression)
= 1 + log2(10) (using the logarithmic identity)
3. log1.6 = log2(1.6) / log2(10) (applying the base-change rule)
≈ 0.737 / log2(10) (calculating the logarithmic value of 1.6)
Now, we substitute these simplified expressions back into the main expression:
3log2 + log20 - log1.6 = 3 + 1 + log2(10) - 0.737 / log2(10)
At this point, we cannot simplify this expression further without additional information or a more specific value for log2(10).
To simplify the expression 3log2 + log20 - log1.6, we will use the properties of logarithms.
First, recall the logarithmic properties:
1. Logarithm of a product: log(ab) = log(a) + log(b)
2. Logarithm of a quotient: log(a/b) = log(a) - log(b)
3. Logarithm of a power: log(a^n) = n * log(a)
Let's simplify each term step by step.
The first term is 3log2. Using property 3, we can rewrite it as log(2^3), which simplifies to log(8).
The second term is log20. Similarly, using property 2, we can rewrite it as log(20/1), which simplifies to log(20).
The last term is log1.6. As there is no operation involved, we can leave it as it is.
Now, let's combine the terms:
log(8) + log(20) - log(1.6)
We can use property 1 to combine the logarithms:
log(8 * 20) - log(1.6)
Simplifying further:
log(160) - log(1.6)
Using property 2 again:
log(160/1.6)
Which simplifies to:
log(100)
Finally, log(100) is equal to 2.
Therefore, the simplified form of 3log2 + log20 - log1.6 is 2.