Question

Logarithms- How to answer these questions?

Answer 1

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

Answer 2

Answer:

For #3: $\log_px=-4$

Step-by-step explanation:

I'm a little rusty on my logarithm rules for #2, but here's an explanation of #3.

Logarithms: the Inverse of Exponents

In a sense, we can think of operations like subtraction and division as different ways of representing addition and multiplication. For instance, the same relationship described by the equation 2 + 3 = 5 is captured in the equation 5 - 3 = 2, and 5 × 2 = 10 can be restated as 10 ÷ 2 = 5 without any loss of meaning.

Logarithms do the same thing for exponents: the expression $2^3=8$ can be expressed in logarithms as $\log_28=3$. Put another way, logarithms are a sort of way of pulling an exponent out onto its own side of the equals sign.

The Problem

Our problem gives us two facts to start: that $log_p(x^2y^3)=7$ and $p^5=y$. With that, we're expected to find the value of $\log_px$. $p^5=y$ stands out as the odd-equation-out here; it's the only one not in terms of logarithms. We can fix that by rewriting it as the equivalent statement $log_py=5$. Now, let's unpack that first logarithm.

Justifying Some Logarithm Rules

For a refresher, let's talk about some of the rules logarithms follow and why they follow them:

Product Rule: $\log_b(MN)=\log_bM+\log_bN$

The product rule turns multiplication in the argument (parentheses) of a logarithm into addition. For a proof of this, consider two numbers $M=b^x$ and $N=b^y$. We could rewrite these two equations with logarithms as $\log_bM=x$ and $\log_bN=y$. With those in mind, we could say the following:

• $\log_b(MN)=\log_b(b^xb^y)$ (Substitution)
• $\log_b(b^xb^y)=log_b(b^{x+y})$ (Laws of exponents)
• $\log_b(b^{x+y})=x+y$ ($\log_b(b^n)=n$)
• $x+y=\log_bM+\log_bN$ (Substitution)

And we have our proof.

Exponent Rule: $\log_b(M^n)=n\log_bM$

Since exponents can be thought of as abbreviations for repeated multiplication, we can rewrite $\log_b(M^n)$ as $\log_b(M\times M\cdots \times M)$, where M is being multiplied by itself n times. From there, we can use the product rule to rewrite our logarithm as the sum $\log_bM+\log_bM+\cdots+\log_bM$, and since we have the term $\log_bM$ added n times, we can rewrite is as $n\log_bM$, proving the rule.

Solving the Problem

With those rules in hand, we're ready to solve the problem. Looking at the equation $\log_p(x^2y^3)=7$, we can use the product rule to split the logarithm into the sum $\log_p(x^2)+\log_p(y^3)=7$, and then use the product rule to turn the exponents in each logarithm's argument into coefficients, giving the equation $2\log_px+3\log_py=7$.

Remember how earlier we rewrote $p^5=y$ as $log_py=5$? We can now use that fact to substitute 5 in for $log_py$, giving us

$2\log_px+3(5)=7$

From here, we can simply solve the equation for $\log_px$:

$2\log_px+15=7\\2\log_px=-8\\\\\log_px=-4$