Solving Exponential Equations: Finding the Value of ‘m’
Exponential equations can appear daunting at first glance, but with a systematic approach and a grasp of the fundamental principles, even complex equations can be solved with ease. In this detailed and well-documented article, we will delve into the step-by-step process of solving an exponential equation of the form 7^m = 70 to find the value of ‘m.’
Step 1: Understanding the Equation We begin with the equation 7^m = 70. Our objective is to determine the value of ‘m’ that satisfies this equation. To simplify this equation, we can express 70 as the product of two numbers: 7 and 10.
Step 2: Splitting the Equation By rewriting 70 as 7 * 10, we can transform our equation into the form 7^m = 7 * 10.
Step 3: Divide by 7 To isolate the exponential term, we divide both sides of the equation by 7. This operation simplifies the equation, and we are left with 7^(m-1) = 10.
Step 4: Applying the Laws of Exponents Here, we can apply the law of exponents, which states that if you have a^(X) / a^(Y), it is equivalent to a^(X-Y). In our equation, this means that 7^(m-1) is equal to 10.
Step 5: Taking the Logarithm To further solve for ‘m,’ we can take the logarithm of both sides of the equation. We choose the base 10 logarithm for convenience.
log(7^(m-1)) = log(10)
Step 6: Using Logarithm Properties Using the properties of logarithms, we can move the exponent (m-1) to the front of the logarithm on the left side:
(m-1) * log(7) = log(10)
Step 7: Isolating ‘m’ Our goal is to find the value of ‘m.’ To do this, we need to isolate ‘m’ on the left side of the equation. We can achieve this by dividing both sides by log(7):
(m-1) = log(10) / log(7)
Step 8: Solving for ‘m’ Now, let’s solve for ‘m’ by adding 1 to both sides of the equation:
m = 1 + (log(10) / log(7))
Step 9: Calculate the Value of ‘m’ Using a calculator to evaluate the logarithms, we find:
m ≈ 1 + (1 / 0.8451)
m ≈ 1 + 1.1842
m ≈ 2.1842
Conclusion After performing the calculations, we find that ‘m’ is approximately equal to 2.1842. This is the solution to the exponential equation 7^m = 70. You can use a calculator to simplify the logarithmic expressions and verify this approximate value in your specific case. Exponential equations may seem complex, but by breaking them down into manageable steps and applying mathematical principles, we can arrive at accurate solutions.