Logarithms
3^x-2=18
Work
log(3^x-2)=log18
(x-2)log3=log18
xlog3-2log3=log18
x=log3/(log9-log18)
e^3x+1=10
lne^3x+1=ln10
3x+1=ln10
I'm stuck here
8^x=5^2x-1
xlog8=(2x-1)log5
xlog8=2xlog5-log5
x=log10/(log5-log8)
Are these right?
humm
3^(x-2) = 18 ? = 2 * 9
3^(x-2) = 2 * 3^2
2 = 3^(x-4)
log 2 = (x-4) log 3
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e^(3x+1) = 10 ?
3x + 1 = ln 10
3x = ln 10 - 1
x = (1/3)ln 10 - 1/3
======================
xlog 8 = x log 2^3 = 3 x log 2
3 x log 2 = 2 x log 5 - log 5
log 5 = x (2 log 5 -3 log 2)
x = log 5/(2 log 5 -3 log 2)
Let's go through each equation step by step to determine if your work is correct.
1. 3^x-2 = 18:
To solve this equation, you correctly took the logarithm of both sides:
log(3^x-2) = log(18)
You then correctly used the logarithmic property:
(x-2)log(3) = log(18)
And finally, you solved for x:
x = log(18) / log(3)
So yes, your work and solution for this equation are correct.
2. e^3x+1 = 10:
To start solving this equation, you took the natural logarithm of both sides:
ln(e^3x+1) = ln(10)
However, you made a mistake in simplifying the left side. The natural logarithm of e raised to any power is equal to that power. So, the correct simplification is:
3x + 1 = ln(10)
Now, you can solve for x by moving the constants to the other side:
3x = ln(10) - 1
x = (ln(10) - 1) / 3
Therefore, your solution for this equation is:
x = (ln(10) - 1) / 3
3. 8^x = 5^(2x-1):
You started solving this equation by taking the logarithm of both sides:
log(8^x) = log(5^(2x-1))
However, you made a mistake in simplifying the left side of the equation. The logarithm of a number raised to a power can be rewritten as the exponent multiplied by the logarithm of that number. So, the correct simplification is:
x log(8) = (2x - 1) log(5)
Now, you can solve for x:
x = (2x - 1) log(5) / log(8)
Therefore, your solution for this equation is:
x = (2x - 1) log(5) / log(8)
Overall, you have correctly used logarithms to solve the equations, but there were some minor mistakes in simplifying the expressions. Make sure to be careful with the logarithmic properties and the rules for exponentiation when simplifying equations involving logarithms.