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Math  /  Algebra

QuestionWrite the expression as a single logarithm. 2log⁡by+3(log⁡bx−5log⁡bz)log⁡□(□)\begin{array}{l} 2 \log _{b} y+3\left(\log _{b} x-5 \log _{b} z\right) \\ \log _{\square}(\square) \end{array}

Studdy Solution

STEP 1

1. We are given an expression involving multiple logarithms.
2. The expression can be simplified using logarithmic properties.
3. The goal is to express the entire expression as a single logarithm.

STEP 2

1. Apply the power rule of logarithms.
2. Distribute and simplify the expression.
3. Combine the logarithms into a single logarithm.

STEP 3

Apply the power rule of logarithms, which states that alog⁡bc=log⁡b(ca) a \log_b c = \log_b (c^a) .
For the term 2log⁡by 2 \log_b y , apply the power rule:
2log⁡by=log⁡b(y2) 2 \log_b y = \log_b (y^2)
For the term 3(log⁡bx−5log⁡bz) 3(\log_b x - 5 \log_b z) , distribute the 3:
3log⁡bx−15log⁡bz 3 \log_b x - 15 \log_b z
Now apply the power rule:
3log⁡bx=log⁡b(x3) 3 \log_b x = \log_b (x^3)
15log⁡bz=log⁡b(z15) 15 \log_b z = \log_b (z^{15})

STEP 4

Substitute the simplified terms back into the expression:
log⁡b(y2)+log⁡b(x3)−log⁡b(z15) \log_b (y^2) + \log_b (x^3) - \log_b (z^{15})

STEP 5

Use the properties of logarithms to combine the terms into a single logarithm. The sum of logarithms corresponds to the product of their arguments, and the difference corresponds to division:
log⁡b(y2⋅x3)−log⁡b(z15)=log⁡b(y2⋅x3z15) \log_b (y^2 \cdot x^3) - \log_b (z^{15}) = \log_b \left(\frac{y^2 \cdot x^3}{z^{15}}\right)
The expression as a single logarithm is:
log⁡b(y2⋅x3z15) \log_b \left(\frac{y^2 \cdot x^3}{z^{15}}\right)

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