2^x × 3^x = 5^(5 + 1)
To solve the equation 2^x × 3^x = 5^(5 + 1), we can rewrite the right side using the properties of exponents and combine the exponents on the same base on the left side.
We know that 5^(5 + 1) is equal to 5^5 × 5^1, which simplifies to 5^6.
Now we have:
2^x × 3^x = 5^6
Since the left side has two different bases (2 and 3), it's difficult to directly compare it to the right side which only has one base (5). To solve this, we can rewrite 2 and 3 in terms of 5.
2 can be written as 5^log5(2) and 3 can be written as 5^log5(3). Substituting these into the equation:
(5^log5(2))^x × (5^log5(3))^x = 5^6
Using the properties of exponents, we can simplify this to:
5^(x * log5(2)) × 5^(x * log5(3)) = 5^6
Now that the bases are the same, we can equate the exponents:
x * log5(2) + x * log5(3) = 6
We can factor out x:
x * (log5(2) + log5(3)) = 6
Now, divide both sides by (log5(2) + log5(3)):
x = 6 / (log5(2) + log5(3))
This is the approximate value of x that satisfies the equation 2^x × 3^x = 5^(5 + 1).
To solve the equation 2^x * 3^x = 5^(5 + 1), we can use logarithms.
Step 1: Take the logarithm of both sides using the same base. We can use the natural logarithm (ln) or the common logarithm (log). Let's use the natural logarithm here for simplicity.
ln(2^x * 3^x) = ln(5^(5 + 1))
Step 2: Apply the logarithmic property that states ln(a * b) = ln(a) + ln(b) to split the left side into two separate logarithms.
ln(2^x) + ln(3^x) = ln(5^(5 + 1))
Step 3: Apply the power rule of logarithms that states ln(a^b) = b * ln(a) to simplify the logarithms on both sides of the equation.
x * ln(2) + x * ln(3) = (5 + 1) * ln(5)
Step 4: Combine like terms on the left side of the equation.
x * (ln(2) + ln(3)) = 6 * ln(5)
Step 5: Divide both sides of the equation by the sum of the natural logarithms of 2 and 3.
x = (6 * ln(5)) / (ln(2) + ln(3))
This is the approximate value of x that satisfies the equation 2^x * 3^x = 5^(5 + 1).