Math Problem Statement

Lab Worksheet 0 (2).pdf

73.65 KB

solve question 3

Solution

The given equations in question 3 are:

apq=btrap^q = bt^r

bsq=atb^s q = at

where aa and bb are constants. We need to express qq as a function of pp.

Step 1: Express qq in Terms of Other Variables

Taking the natural logarithm on both sides of the first equation:

ln(apq)=ln(btr)\ln(ap^q) = \ln(bt^r)

Using logarithm properties:

lna+qlnp=lnb+rlnt\ln a + q \ln p = \ln b + r \ln t

Rearrange for qq:

q=lnb+rlntlnalnpq = \frac{\ln b + r \ln t - \ln a}{\ln p}

Step 2: Express tt in Terms of Other Variables

From the second equation:

bsq=atb^s q = at

Solving for tt:

t=bsqat = \frac{b^s q}{a}

Step 3: Substitute tt into the Expression for qq

Substituting t=bsqat = \frac{b^s q}{a} into the equation for qq:

q=lnb+rln(bsqa)lnalnpq = \frac{\ln b + r \ln \left(\frac{b^s q}{a} \right) - \ln a}{\ln p}

Expanding the logarithm:

q=lnb+r(lnbs+lnqlna)lnalnpq = \frac{\ln b + r (\ln b^s + \ln q - \ln a) - \ln a}{\ln p}

q=lnb+r(slnb+lnqlna)lnalnpq = \frac{\ln b + r (s \ln b + \ln q - \ln a) - \ln a}{\ln p}

q=lnb+rslnb+rlnqrlnalnalnpq = \frac{\ln b + rs \ln b + r \ln q - r \ln a - \ln a}{\ln p}

Step 4: Solve for qq

Rearrange to isolate qq:

qrlnqlnp=lnb+rslnb(r+1)lnalnpq - \frac{r \ln q}{\ln p} = \frac{\ln b + rs \ln b - (r+1) \ln a}{\ln p}

Factor qq out:

q(1rlnp)=lnb+rslnb(r+1)lnalnpq \left( 1 - \frac{r}{\ln p} \right) = \frac{\ln b + rs \ln b - (r+1) \ln a}{\ln p}

Solving for qq:

q=lnb+rslnb(r+1)lnalnprq = \frac{\ln b + rs \ln b - (r+1) \ln a}{\ln p - r}

Thus, the function for qq in terms of pp is:

q(p)=lnb+rslnb(r+1)lnalnprq(p) = \frac{\ln b + rs \ln b - (r+1) \ln a}{\ln p - r}

Would you like a detailed numerical example to illustrate this?

Related Questions:

  1. How would the solution change if s=1s = 1 and r=2r = 2?
  2. What happens to qq if p1p \to 1?
  3. How can logarithmic identities be used to simplify this result further?
  4. What happens if b=1b = 1 in the given equations?
  5. Can this equation be solved graphically for qq as a function of pp?

Tip:

When solving logarithmic equations, always check for domain restrictions to ensure valid solutions!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logarithms
Exponential Equations
Algebraic Manipulation

Formulas

ln(ap^q) = ln(bt^r)
ln a + q ln p = ln b + r ln t
b^s q = at

Theorems

Logarithmic Properties
Exponentiation Rules

Suitable Grade Level

Undergraduate Mathematics / Advanced High School