Answer:
2. m = b³ (= 216)
3. logp(x) = -4
Step-by-step explanation:
2. The given equation can be written using the change of base formula as ...
... log(m)/log(b) + 9·log(b)/log(m) = 6
If we define x = log(m)/log(b), then this becomes ...
... x + 9/x = 6
Subtracting 6 and multiplying by x gives ...
... x² -6x +9 = 0
... (x -3)² = 0 . . . . . factored
... x = 3 . . . . . . . . . value of x that makes it true
Remembering that x = log(m)/log(b), this means
... 3 = log(m)/log(b)
... 3·log(b) = log(m) . . . . . multiply by the denominator; next, take the antilog
... m = b³ . . . . . . the expression you're looking for
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3. Substituting the given expression for y, the equation becomes ...
... logp(x^2·(p^5)^3) = 7
... logp(x^2) + logp(p^15) = 7 . . . . . use the rule for log of a product
... 2logp(x) + 15 = 7 . . . . . . . . . . . . . use the definition of a logarithm
... 2logp(x) = -8 . . . . . . . . . . . . . . . . subtract 15
... logp(x) = -4 . . . . . . divide by 2
