solve each equation 1) log5-log2x=1 2) 2log(x+1)=5
1)
log5 - log(2x) = 1
log(5/2x) = 1
5/(2x) = 10^1
20x = 5
x = 1/4
2)
2log(x+1) = 5
log(x+1) = 5/2
10^(5/2) = x+1
316.227766 = x+1
x = 315.227766
OR
log(x+1)^2 = 5
(x+1)^2 = 10^5
x+1 = ±√(10^5) , but clearly x >-1
x+1 = 316.227766..
x = 315.227766..
1)> log 5 + log 2 + log x = 1
>log (5 X 2) + log x = 1
>1 + log x = 1
>log x = 0
>x = 10^0
>x = 1
#2 has me twisted.
To solve each equation step-by-step:
1) log5 - log2x = 1
We can start by combining the logarithms using one of the logarithmic identities, which states that log(a) - log(b) = log(a/b).
So, we can rewrite the equation as:
log(5/2x) = 1
Next, we can convert the logarithmic equation into an exponential equation. The logarithmic equation log(base a)(x) = b is equivalent to the exponential equation a^b = x.
Using this, we can rewrite the equation as:
5/2x = 10^1
Simplifying further:
5/2x = 10
To solve for x, we can cross-multiply:
5 = 2x * 10
Dividing both sides by 2 gives us:
x = 5/20
Simplifying:
x = 1/4
So, the solution to the equation is x = 1/4.
2) 2log(x+1) = 5
We can start by dividing both sides of the equation by 2:
log(x+1) = 5/2
Next, we can convert the logarithmic equation into an exponential equation:
x+1 = 10^(5/2)
Taking the square root of both sides:
√(x+1) = √(10^(5/2))
Simplifying:
√(x+1) = √(√(100) * √(10))
√(x+1) = √(10 * √(10))
√(x+1) = √(10) * √(√(10))
√(x+1) = √(10) * √(√(√(100)))
√(x+1) = √(10) * √(2)
Squaring both sides:
x+1 = 10 * 2
x+1 = 20
Subtracting 1 from both sides:
x = 20 - 1
x = 19
So, the solution to the equation is x = 19.
To solve each equation, we'll follow the steps and properties of logarithms.
1) log(5) - log(2x) = 1
Step 1: Combine the logarithms using the quotient rule of logarithms. According to the quotient rule, log(a) - log(b) can be rewritten as log(a / b).
log(5) - log(2x) = log(5 / 2x) = 1
Step 2: Rewrite the equation in exponential form. In logarithmic form, log(base)(value) = exponent. In exponential form, base^(exponent) = value.
5 / 2x = 10^1
Step 3: Simplify the right-hand side.
5 / 2x = 10
Step 4: Solve for x.
Cross-multiply to eliminate the fraction:
5 = 10 * 2x
5 = 20x
Divide both sides by 20 to solve for x:
x = 5 / 20
Simplifying further:
x = 1 / 4
Therefore, the solution to the equation is x = 1/4.
2) 2log(x+1) = 5
Step 1: Divide both sides of the equation by 2 to isolate the logarithm.
log(x+1) = 5/2
Step 2: Convert the logarithmic equation into exponential form.
10^(log(x+1)) = 10^(5/2)
(x+1) = √(10^5)
Step 3: Evaluate the right-hand side.
(x+1) = √(100,000)
Step 4: Simplify the square root expression.
(x+1) = 316.2278
Step 5: Solve for x.
x = 316.2278 - 1
x = 315.2278
Therefore, the solution to the equation is x = 315.2278.