10x^2=2^x
Hi!
How do I go about solving an equation like this? It's meant to be a calculator solution but I'd love to know how it's done by hand. Logarithms maybe? .. Thank you in advance (:
10 x^2 = 2^x
ah
well, we will need the calculator or table to do the logs anyway but
log (10 x^2) = log 10 + 2 log x
if base ten logs
= 1 + 2 log x
log 2^x = x log 2
so
1 + 2 log x = x log 2
I need my slide rule or a calculator now :)
slide rule ?
Damon, you must be almost as old as I am.
Wolfram shows 3 real solutions
http://www.wolframalpha.com/input/?i=solve+10x%5E2%3D2%5Ex
Hello! To solve the equation 10x^2 = 2^x by hand, we can indeed use logarithms. Specifically, we'll use logarithmic functions to isolate the x term.
Here's the step-by-step process:
Step 1: Take the logarithm of both sides of the equation. The choice of logarithm base is up to you, but it's commonly done with natural logarithm (logarithm base e) or common logarithm (logarithm base 10).
For this example, let's use natural logarithm (ln):
ln(10x^2) = ln(2^x)
Step 2: Apply the logarithm rule to bring down the exponent in the equation. The rule states that the logarithm of a number raised to a power is equal to the product of the exponent and the logarithm of the number itself. In other words, it allows us to bring the exponent down as a coefficient:
2xln(10) = xln(2)
Step 3: Now, rearrange the equation to isolate the x term. You can subtract xln(2) from both sides of the equation:
2xln(10) - xln(2) = 0
Step 4: Factor out the common x term on the left side:
x(2ln(10) - ln(2)) = 0
Step 5: Next, solve for x by setting each factor equal to zero. This means that either x = 0 or (2ln(10) - ln(2)) = 0. Solving the second factor:
2ln(10) - ln(2) = 0
Step 6: Combine the logarithms using the properties of logarithms. In this case, we can combine the two logarithms as follows:
ln(10^2) - ln(2) = 0
ln(100) - ln(2) = 0
ln(100/2) = 0
ln(50) = 0
Step 7: Finally, solve for x. Using a calculator to evaluate ln(50), we find that x ≈ 3.912023.
So, the solution to the equation 10x^2 = 2^x is approximately x ≈ 3.912023.
I hope this explanation helps you understand how to solve this equation by hand! Let me know if you have any further questions.