solve each equation algebraically and check it by substituing into the orignall equation.
50e^0.035x=200
3LN(x-3)+4=5
Method of your choice by solving.
logx^2=6
Logx^4=2
2x-2^-x/2=4
e^x+e^-x/2=4
500/1+25e^.3x=200
50e^(0.035x)=200
e^(.035x) = 4
take ln of both sides
ln(e^(.035x)) = ln4
.035x lne = ln4, but lne = 1
x = ln4 / .035 = appr. 39.6
log x^2 = 6
by definition:
x^2 = 10^6
x = 10^3 = 1000
do logx^4 = 2 the same way
2x-2^-x/2=4 : I have a feeling that is not what you meant, without brackets I cannot tell what the equation is. I think the first term is probably 2^x
e^x+e^-x/2=4
Again, I think you meant:
e^x+e^(-x/2) = 4
let e^(-x/2) = y , where y is a positive real number
then e^(x/2) = 1/y
and e^x = 1/y^2
so 1/y2 + y = 4
1 + y^3 = 4y^2
y^3 - 4y^2 + 1 = 0
I used a program to find
y = .54 or y = 3.94 appr. , there is also a negative root which would not be allowed
so e^(-x/2) = .54 or ..... use the other root
-x/2 = ln.54
x = 1.2324 or ......
last question, way too ambiguous without brackets.
To solve each equation algebraically and check it by substituting into the original equation, we can follow these steps:
1. 50e^(0.035x) = 200
To solve this equation, we can start by dividing both sides by 50:
e^(0.035x) = 4
Take the natural logarithm (ln) of both sides:
ln(e^(0.035x)) = ln(4)
Using the property that ln(e^a) = a, we can simplify the left side:
0.035x = ln(4)
Now, divide both sides by 0.035 to isolate x:
x = ln(4) / 0.035
To check the solution, substitute the obtained value of x back into the original equation:
50e^(0.035 * (ln(4) / 0.035)) = 200
Simplify and check if both sides are equal.
2. 3ln(x - 3) + 4 = 5
To solve this equation, we'll first isolate the natural logarithm term:
3ln(x - 3) = 5 - 4
3ln(x - 3) = 1
Divide both sides by 3:
ln(x - 3) = 1/3
Take the exponential (e) of both sides to eliminate the logarithm:
e^(ln(x - 3)) = e^(1/3)
Simplify the left side:
x - 3 = e^(1/3)
Now, isolate x by adding 3 to both sides of the equation:
x = e^(1/3) + 3
To check the solution, substitute the obtained value of x back into the original equation:
3ln((e^(1/3) + 3) - 3) + 4 = 5
Simplify and check if both sides are equal.
3. log(x^2) = 6
To solve this equation, rewrite it in exponential form:
x^2 = 10^6
Simplify the right side:
x^2 = 1,000,000
Take the square root of both sides:
x = ±√(1,000,000)
Simplify the square root:
x = ±1000
To check the solution, substitute the obtained values of x back into the original equation:
log((1000)^2) = 6
log(1,000,000) = 6
Simplify and check if both sides are equal.
4. log(x^4) = 2
To solve this equation, rewrite it in exponential form:
x^4 = 10^2
Simplify the right side:
x^4 = 100
Take the fourth root of both sides:
x = ±√(√100)
Simplify the fourth root:
x = ±√(10)
To check the solution, substitute the obtained values of x back into the original equation:
log((√10)^4) = 2
log(10) = 2
Simplify and check if both sides are equal.
5. 2x - 2^(-x/2) = 4
To solve this equation, we'll isolate the exponential term:
2^(-x/2) = 2x - 4
Take the logarithm (base 2) of both sides to eliminate the exponential term:
log2(2^(-x/2)) = log2(2x - 4)
Using the logarithmic property loga(b^c) = cloga(b), we can simplify the left side:
(-x/2)log2(2) = log2(2x - 4)
Since log2(2) = 1, the equation becomes:
(-x/2)(1) = log2(2x - 4)
Simplify the left side:
-x/2 = log2(2x - 4)
Multiply both sides by -2 to isolate x:
x = -2log2(2x - 4)
To check the solution, substitute the obtained value of x back into the original equation:
2(-2log2(2x - 4)) - 2^(-(-2log2(2x - 4))/2) = 4
Simplify and check if both sides are equal.
6. e^x + e^(-x/2) = 4
To solve this equation, we'll isolate the exponential term:
e^(-x/2) = 4 - e^x
Multiply both sides by e^(x/2) to eliminate the exponential term:
1 = (4 - e^x)e^(x/2)
Expand the right side:
1 = 4e^(x/2) - e^x
Rearrange the equation:
e^x - 4e^(x/2) + 1 = 0
This equation does not have a simple algebraic solution, so we can use numerical methods or graphical analysis to approximate the solution.
7. 500 / (1 + 25e^(0.3x)) = 200
To solve this equation, we'll start by isolating the denominator:
1 + 25e^(0.3x) = 500 / 200
Simplify the right side:
1 + 25e^(0.3x) = 5/2
Subtract 1 from both sides:
25e^(0.3x) = 5/2 - 1
Simplify the right side:
25e^(0.3x) = 3/2
Divide both sides by 25:
e^(0.3x) = 3/50
Take the natural logarithm (ln) of both sides:
ln(e^(0.3x)) = ln(3/50)
Using the property that ln(e^a) = a, we can simplify the left side:
0.3x = ln(3/50)
Now, divide both sides by 0.3 to isolate x:
x = ln(3/50) / 0.3
To check the solution, substitute the obtained value of x back into the original equation:
500 / (1 + 25e^(0.3 * (ln(3/50) / 0.3))) = 200
Simplify and check if both sides are equal.