These are Two problems that I am having difficulty with.
Write as a single logarithm. Assume all variables represent positive real numbers.
1. 5log3 a-7 log3 b+2 log3c
2.log7(x2+5x-24)-log7(x+8)
log a^3 - log b^7 + log c^2
= log3(a^3 c^2/b^7 )
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x^2+5x-24 = (x+8)(x-3)
so
log7 (x-3)
Sure! I can help you with those problems. To simplify these expressions, we will use logarithmic properties to combine the logarithms into a single logarithm. Here's how you can solve each problem step-by-step:
Problem 1: Write as a single logarithm.
Given expression: 5log3 a - 7log3 b + 2log3 c
To combine these logarithms, we will use the following properties:
1. The power rule: log base a (x^m) = m * log base a (x)
2. The product rule: log base a (x * y) = log base a (x) + log base a (y)
3. The quotient rule: log base a (x / y) = log base a (x) - log base a (y)
Using these properties, let's simplify the expression:
5log3 a - 7log3 b + 2log3 c
= log3 a^5 - log3 b^7 + log3 c^2 (Applying the power rule)
Now, we can use the product rule to combine these logarithms:
= log3 (a^5 * c^2) - log3 b^7
Finally, we have a single logarithm:
= log3 (a^5 * c^2 / b^7)
Answer: The expression can be written as log base 3 of (a^5 * c^2 / b^7).
Moving on to the second problem:
Problem 2: Write as a single logarithm.
Given expression: log7(x^2 + 5x - 24) - log7(x + 8)
To combine these logarithms, we will use the quotient rule.
Using the quotient rule, we have:
= log7((x^2 + 5x - 24) / (x + 8))
Now, simplify the numerator and the denominator:
= log7((x - 3)(x + 8) / (x + 8))
Since (x + 8) appears in both the numerator and the denominator, we can cancel it out:
= log7(x - 3)
Answer: The expression can be written as log base 7 of (x - 3).
I hope this helps! Let me know if you have any other questions.