https://file.aiquickdraw.com/mathbot/file/1726125280878sym8skhk.jpg

Let's go through each equation and make the variable in brackets the subject of the formula.

### a) \( A = \frac{a}{b} \) (Make \( a \) the subject)
Multiply both sides by \( b \):
\[
a = A \cdot b
\]

### b) \( \log_{10} \left( \frac{a}{x} \right) = k \) (Make \( x \) the subject)
1. Raise both sides as powers of 10 to eliminate the logarithm:
   \[
   \frac{a}{x} = 10^k
   \]
2. Multiply both sides by \( x \):
   \[
   a = x \cdot 10^k
   \]
3. Divide both sides by \( 10^k \):
   \[
   x = \frac{a}{10^k}
   \]

### c) \( ke^{at+b} = e^t \) (Make \( t \) the subject)
1. Divide both sides by \( e^t \):
   \[
   k e^{at + b - t} = 1
   \]
2. Simplify the exponent:
   \[
   k e^{(a-1)t + b} = 1
   \]
3. Take the natural logarithm (ln) of both sides:
   \[
   (a-1)t + b = \ln(1) - \ln(k) = - \ln(k)
   \]
4. Solve for \( t \):
   \[
   (a-1)t = -\ln(k) - b
   \]
   \[
   t = \frac{-\ln(k) - b}{a - 1}
   \]

### d) \( t = \frac{1}{k} \log_e (1 - \frac{x}{y}) \) (Make \( x \) the subject)
1. Multiply both sides by \( k \):
   \[
   kt = \log_e \left( 1 - \frac{x}{y} \right)
   \]
2. Raise both sides as powers of \( e \):
   \[
   e^{kt} = 1 - \frac{x}{y}
   \]
3. Subtract \( e^{kt} \) from both sides:
   \[
   \frac{x}{y} = 1 - e^{kt}
   \]
4. Multiply both sides by \( y \):
   \[
   x = y(1 - e^{kt})
   \]

Let me know if you'd like any further clarifications!

### Related Questions:
1. How do you solve logarithmic equations in general?
2. What are the properties of logarithms that simplify calculations?
3. How do exponential equations relate to logarithmic ones?
4. What are some applications of natural logarithms in real-life problems?
5. How can logarithmic and exponential equations be used in solving growth/decay models?

### Tip:
When dealing with logarithmic equations, remember that the inverse of a logarithm is an exponential function. This helps in eliminating the logarithmic term by exponentiation.

32) Make the variables in brackets subject of the formulas: a) A = a/b, b) log10(a/x) = k, c) ke^(at+b) = e^t, d) t = (1/k)log_e(1 - (x/y))

Learn how to make variables the subject in logarithmic and exponential equations with detailed steps. This guide covers key algebraic formulas such as A = a/b, log10(a/x) = k, ke^(at+b) = e^t, and t = (1/k)log_e(1 - (x/y)). Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation