Math Problem Statement

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-cke-white-space-first="1" data-cke-white-space-last="1"> <mo>log</mo> <mfenced close=")" open="("> <mrow> <mi mathvariant="italic">x</mi> </mrow> </mfenced> <mo></mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>log</mo> <mfenced close=")" open="("> <mrow> <mi mathvariant="italic">y</mi> </mrow> </mfenced> <mo>+</mo> <mn>7</mn> <mo>log</mo> <mfenced close=")" open="("> <mrow> <mi mathvariant="italic">z</mi> </mrow> </mfenced><mo>=</mo> </mrow></math>

Solution

We are tasked with simplifying or solving the logarithmic equation:

log(x)12log(y)+7log(z)=0\log(x) - \frac{1}{2} \log(y) + 7 \log(z) = 0

Step-by-step explanation:

  1. Apply the logarithmic property alog(b)=log(ba)a \log(b) = \log(b^a):

    • For 12log(y)-\frac{1}{2} \log(y), we rewrite it as: log(y12)=log(1y)\log(y^{-\frac{1}{2}}) = \log\left(\frac{1}{\sqrt{y}}\right)
    • For 7log(z)7 \log(z), we rewrite it as: log(z7)\log(z^7)

    Substituting these into the equation, we get: log(x)+log(1y)+log(z7)=0\log(x) + \log\left(\frac{1}{\sqrt{y}}\right) + \log(z^7) = 0

  2. Combine the logarithms using the property log(a)+log(b)=log(ab)\log(a) + \log(b) = \log(a \cdot b):

    • Combine all terms into a single logarithm: log(x1yz7)=0\log\left(x \cdot \frac{1}{\sqrt{y}} \cdot z^7\right) = 0

    Simplify further: log(xz7y)=0\log\left(\frac{x z^7}{\sqrt{y}}\right) = 0

  3. Exponentiate to remove the logarithm:

    Recall that log(a)=b    a=10b\log(a) = b \implies a = 10^b. Therefore: xz7y=100\frac{x z^7}{\sqrt{y}} = 10^0

    Since 100=110^0 = 1, we have: xz7y=1\frac{x z^7}{\sqrt{y}} = 1

  4. Solve for xx:

    Multiply through by y\sqrt{y}: xz7=yx z^7 = \sqrt{y}

    Finally: x=yz7x = \frac{\sqrt{y}}{z^7}

Final Simplified Expression:

x=yz7x = \frac{\sqrt{y}}{z^7}

Would you like further clarification or examples on logarithmic properties?


Related Questions:

  1. What are the laws of logarithms and how are they applied in equations?
  2. How do you handle equations involving logarithms with different bases?
  3. What is the geometric interpretation of logarithmic equations like this one?
  4. How can we extend this method to solve systems of logarithmic equations?
  5. What are common mistakes to avoid when simplifying logarithmic expressions?

Tip:

When solving logarithmic equations, always check for domain restrictions to ensure all variables meet the conditions x>0x > 0, y>0y > 0, and z>0z > 0.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Equations
Properties of Logarithms
Simplification of Expressions

Formulas

a * log(b) = log(b^a)
log(a) + log(b) = log(a * b)
log(a) = b implies a = 10^b

Theorems

Properties of Logarithms
Exponentiation to Remove Logarithms

Suitable Grade Level

Grades 10-12