This is for solving exponential/logarithmic functions:
(This is a base e Logarithmic function I would assume):
e^(4x)/10 =4^x-2 ?
I understand the properties of logs for the most part, but I have a hard time figuring out the step-by-step process on how to solve exponential/log equations? Is your first step always to take the ln of both sides, or is that only for certain types of equations? I need a a step to step list on how to work this, to where I can understand and it's just not all math book definitions. For example, I tried to write out my own process such as:
log equations:
1. take log of both sides
2. drop logs
3. multiply
4. distribute
5. standard form
6. solve
Would this be correct?
for exponentials,
1. collect exponent stuff on one side
2. take logs
add/subtract terms to get x's on one side
divide by coefficient to get x alone
for the above one,
e^(4x)/10 =4^x-2
e^(4x) = 10*4^x - 20
not much you can do now. That pesky -20 gets in the way.
To solve the given exponential/logarithmic equation e^(4x)/10 = 4^(x-2), you can follow these step-by-step instructions:
1. Take the natural logarithm (ln) of both sides of the equation. This step is generally taken when dealing with exponential equations involving e, as ln is the inverse function of e and helps simplify the equation.
ln(e^(4x)/10) = ln(4^(x-2))
2. Use logarithmic properties to simplify the equation. The first property states that ln(a/b) = ln(a) - ln(b), and the second property states that ln(a^b) = b * ln(a).
ln(e^(4x)) - ln(10) = (x-2) * ln(4)
Using the second property, ln(e^(4x)) simplifies to 4x * ln(e), and ln(4) can be treated as a constant.
4x * ln(e) - ln(10) = (x-2) * ln(4)
3. Simplify the equation further. Recall that ln(e) is equal to 1, so the term 4x * ln(e) simplifies to just 4x.
4x - ln(10) = (x-2) * ln(4)
4. Distribute the ln(4) term on the right side of the equation:
4x - ln(10) = x * ln(4) - 2 * ln(4)
5. Collect like terms on each side of the equation. Move the x term to one side and the constant terms to the other side.
4x - x * ln(4) = - 2 * ln(4) + ln(10)
x(4 - ln(4)) = -2 * ln(4) + ln(10)
6. Divide both sides of the equation by (4 - ln(4)) to solve for x.
x = (-2 * ln(4) + ln(10))/(4 - ln(4))
7. Use a calculator to determine the decimal approximation of x.
Note: The steps provided here are applicable for this specific equation. Generally, the process of solving exponential/logarithmic equations may vary depending on the given equation.
To solve exponential and logarithmic equations, the approach depends on the specific equation. However, I can provide you with a general step-by-step process that can be helpful in most cases.
Let's work with the equation you provided as an example: e^(4x)/10 = 4^(x-2).
Step 1: Isolate the exponential or logarithmic term:
To start, we want to bring either the exponential term or the logarithmic term by itself on one side of the equation. In this case, let's isolate the exponential term by multiplying both sides of the equation by 10:
e^(4x) = 10 * 4^(x-2).
Step 2: Simplify the equation:
Now, we want to simplify the equation as much as possible. In this case, we can simplify the right side by expressing 4^(x-2) as (2^2)^(x-2):
e^(4x) = 10 * 2^(2x-4).
Step 3: Convert the exponential equation to logarithmic or vice versa:
With exponential equations, we can often use logarithms to solve them. In this case, taking the natural logarithm (ln) of both sides of the equation can help:
ln(e^(4x)) = ln(10 * 2^(2x-4)).
Using the property ln(e^a) = a, the equation becomes:
4x = ln(10 * 2^(2x-4)).
Step 4: Simplify further if possible:
If there are any simplifications that can be made, do so at this point. For example, you can distribute the ln to both terms inside the parentheses:
4x = ln(10) + ln(2^(2x-4)).
Step 5: Solve for the unknown:
Now, we can solve for x. In this case, it might be helpful to isolate the term involving x first. Let's subtract ln(10) from both sides:
4x - ln(10) = ln(2^(2x-4)).
Step 6: Use properties of logarithms to further simplify:
Using the property ln(a^b) = b * ln(a), the equation becomes:
4x - ln(10) = (2x - 4) * ln(2).
Step 7: Solve for x:
From here, you can solve for x by isolating the x term. Let's distribute ln(2) on the right side of the equation:
4x - ln(10) = 2x * ln(2) - 4 * ln(2).
Now, move all the x terms to one side and the non-x terms to the other side:
4x - 2x * ln(2) = -ln(10) - 4 * ln(2).
Finally, divide both sides of the equation by 2 - ln(2) to solve for x:
x = (-ln(10) - 4 * ln(2)) / (4 - 2 * ln(2)).
That's it! You've solved the equation. Remember to double-check your solution and consider any domain restrictions that may be applicable.
I hope this step-by-step process helps you understand how to approach and solve exponential/logarithmic equations.