Given m=log a, n= log b, and k= log ab

Question

a. express a, b, and ab as powers of 10

b. with knowledge of exponents what is the relationship among m, n, and k

c. conclude that log (ab) = log a + log b

Justify rules:

a. log (a^r) = r log a

b. log (a/b) = log a - log b

Answer 1

(a) 10^m = 10^log10(a). The 10^log10 cancels out b/c the definition of logarithmic functions is to the power 10; logarithmic functions are the inverse functions of exponentiation, or exponents, such that:

log_b(b^x) = x.

Given this relationship, we can rearrange variables with values given in the presented question stem whereby a, b, and ab may be expressed as powers of 10. Lets say we set the base of the logarithm, "b" in the image aforementioned. Then log₁₀(10^x) = x, because the " log₁₀(10" term on the left side of the equation cancels out, leaving only the x remaining.

Given that m = log(a), we can presume the base of the logarithmic function to be 10, such that exponentiating both sides of the equation with 10^(***), we can cancel out the logarithmic function (since logs are the inverse functions of exponentiation). The following is what comes from exponentiating both sides by the base factor of 10:

10^m = 10^log₁₀^(a) . The 10log₁₀ of course cancels out, leaving us with 10^m = a.

n = log₁₀(b) same thing leading to 10ⁿ = b.

k = log₁₀(ab) : 10^k = ab.

(b) Knowledge of exponents, the fact that log functions are the inverse of exponentiation functions, and that we can combine terms following simplifications, produces the following proof and derivation:

ab = 10^k = (10^m)(10ⁿ). Because the bases, 10, are the same in each factor undergoing multiplication on the right side of the equation, the exponents may be summed which equates to the terms being multiplied. This leaves us with:

10^k = 10^{m + n}. Given that the bases of 10 are equivalent on both sides of the equation, we can equate the exponents to each other leaving us with:

k = m + n.

(c) Coming full circle, if we take the log of both side of the 10^k = ab function, we get the following:

log₁₀10^k = log₁₀(ab). Simplifying:

k = log₁₀(ab). As we derived up above, k = m + n.

m = log₁₀(a)

n = log₁₀(b).

So, k = log₁₀(ab) = log₁₀(a) + log₁₀(b). This concludes the proof with the final (c) described in the question stem.

Answer 2

a.

Our definition of log tells us that if m = log(a), then 10^m = a. So, a = 10^m, b = 10ⁿ, and ab = 10^k.

b.

With exponent rules, ab = (10^m)(10ⁿ) = 10^m+n = 10^k. For this equation to be true, m+n=k.

c.

If m+n=k, we can use our original log definitions to say that log(a) + log(b) = log(ab).