logarithms?
log (base 9) x+3 log (base 3) x=14
To solve the equation log (base 9) x + 3 log (base 3) x = 14, we can use logarithmic properties and convert all the logarithms to the same base:
1. Start by using the property log (a) + log (b) = log (a * b) to combine the logarithms:
log₉ (x) + log₃ (x³) = 14
2. Next, use the property log (aᵇ) = b * log (a) to simplify the expression:
log₉ (x) + 3 * log₃ (x) = 14
3. Now, convert the logarithms to the same base. Since 3 is a power of 9 (9^2 = 81), we can rewrite log₉ (x) as log₃ (√x) and log₃ (x) as log₉ (x^(1/2)):
log₃ (√x) + 3 * log₉ (x^(1/2)) = 14
4. Use the property logₐ (a) = 1 to simplify further:
log₃ (√x) + 3 * (1/2) * log₉ (x) = 14
5. Simplify the equation:
log₃ (√x) + (3/2) * log₉ (x) = 14
6. Now, rewrite the equation using the property (logₐ (b)) / (logₐ (c)) = log₋ (c) (b) (logarithm change of base formula):
log₉ (√x) / log₉ (3) + (3/2) * log₉ (x) = 14
7. Since logₙ (√a) = (1/2) * logₙ (a), we can simplify further:
(1/2) * log₉ (x) / log₉ (3) + (3/2) * log₉ (x) = 14
8. Combine the terms with the same base:
[ (1/2) / log₉ (3) + (3/2) ] * log₉ (x) = 14
9. Simplify the coefficient:
[ (1/2) + (3/2) ] * log₉ (x) = 14
10. Combine the fractions:
(2/2) * log₉ (x) = 14
11. Simplify further:
log₉ (x) = 14
12. Rewrite the equation exponentiated with base 9:
x = 9^14
Therefore, the solution to the equation log (base 9) x + 3 log (base 3) x = 14 is x = 9^14.
To solve the equation log (base 9) x + 3 log (base 3) x = 14, we can apply the properties of logarithms.
First, let's simplify the equation using the rules of logarithms:
Using the property of logarithms, log (base a) b + log (base a) c = log (base a) (b * c), we can rewrite the equation as:
log₉(x) + log₃(x³) = 14
Now we can use another property of logarithms, logₐ(b) = (logₓ(b))/(logₓ(a)), to change the bases. We will change the bases to logₓ(x) for convenience:
logₓ(x) / logₓ(9) + 3 (logₓ(x) / logₓ(3³)) = 14
Since all the bases are now the same (logₓ), we can now simplify further:
(logₓ(x) + 3 logₓ(x)) / logₓ(9) = 14
Now, we can simplify the numerator:
logₓ(x) + 3 logₓ(x) = 4 logₓ(x)
Substituting these simplified values back into the equation:
(4 logₓ(x)) / logₓ(9) = 14
Now, we can solve for logₓ(x):
4 logₓ(x) = 14 logₓ(9)
Dividing both sides by 4:
logₓ(x) = 3.5 logₓ(9)
To solve for x, we need to eliminate the logarithm. We can do this by applying the exponentiation property:
x = 9^(3.5 logₓ(9))
At this point, we can choose a base for x. Let's choose x = 10 for simplicity:
x = 9^(3.5 log₁₀(9))
Using a calculator to evaluate log₁₀(9), we get:
x ≈ 4183.78
So, the solution to the equation log (base 9) x + 3 log (base 3) x = 14, is approximately x = 4183.78, when x is chosen as the base 10 logarithm.
log9(x) = log3(x) * log9(3)
what is log9(3)
9^log9(3) = 3
9^.5 = 3
so
log 9(x) = .5 log3(x)
Now the problem
.5 log3(x) + 3 log3(x) = 14
log3 (x^.5) + log3(x^3) = 14
log 3 (x^3.5) = 14
x^3.5 = 3^14
now use plain old base ten logs
3.5 log x = 14 log 3
log x = 1.91
x = 10^1.91 = 81.3
Check my arithmetic!!!