My question says:
Find the exact alue of x for which (4^x)*(5^[4x+3])=(10^[2x+3])
I can't seem to come to a solution.
We're reviewing last year's lessons, so change of base and logarithmic expressions are what we're going over right now.
Here's what I've done so far:
(4^x)*(5^[4x+3])=(10^[2x+3])
log both sides
log[(4^x)*(5^[4x+3])]=log[(10^[2x+3])]
(xlog4)+([4x+3]log5)=([2x+3]log10)
since log10=1
(xlog4)+([4x+3]log5)=(2x+3)
Have I started correctly? Where do I go from here? Please be detailed, I want to understand this.
Proceed and solve for x in terms of ln.
I get x=ln(8)/ln(25).
How do I proceed? I really have no clue where to go from where I've stopped! Usually I can simply factor out an x from the LS, but here I can't do that. Could you please show me your solution with every step? It would be so helpful.
Thank you in advance :)
It is a linear equation with numerical coefficients (log4, etc.).
Expand
(xlog4)+([4x+3]log5)=(2x+3)
to give
xlog4 + 4xlog5 - 2x = 3 -3log5
Factor out x on the LHS and solve for x.
All the log4, log5 are numerical values that you can simplify eventually.
Sorry to keep bugging, but I followed what you told me and got to:
x=(3-[log125])/([log2500]-2)
How can I get to the answer from that? I don't know how to deal with the 3 and the -2 that are still lying around.
Alright, what I did now was:
since log10=1, log100=2, and log1000=3
So I plugged that in to what I had
x=(log1000-log125)/(log2500-log100)
x=(log[1000/125])/(log[2500/100])
x=(log8)/(log25)
Is that what I was supposed to do?
Yes, you have started correctly. The next step is to simplify the expression further and solve for x.
To continue, let's simplify the equation using the properties of logarithms and the fact that log10 = 1.
(x log4) + ([4x + 3] log5) = (2x + 3)
Now, let's apply the property of logarithms which states that log(a * b) = log(a) + log(b). We can use this to rewrite the left side of the equation.
x log4 + [log(5^(4x+3))] = 2x + 3
Next, we can use another property of logarithms, which is that log(a^b) = b * log(a). Applying this property to the logarithm term, we get:
x log4 + (4x + 3)(log5) = 2x + 3
Now, let's distribute (log5) to both terms inside the parentheses:
x log4 + 4x * log5 + 3 * log5 = 2x + 3
Next, combine like terms on both sides of the equation. Let's start by collecting all terms with x on one side and all constant terms on the other side.
x log4 - 2x = - 4x * log5 + 3 * log5 - 3
Simplifying further:
x(log4 - 2) = -log5(4x - 3) + 3
Now, let's divide both sides of the equation by (log4 - 2) to isolate x:
x = [ -log5(4x - 3) + 3 ] / (log4 - 2)
At this point, we have an expression for x, but it is not in a simplified form yet. To simplify further, you can distribute -log5 inside the parentheses:
x = [ -(log5 * 4x) + (log5 * 3) + 3 ] / (log4 - 2)
Now, distribute the -log5 to each term inside the parentheses:
x = [ -4x * log5 + 3 * log5 + 3 ] / (log4 - 2)
Finally, you can simplify the expression further by factoring out x from the numerator:
x = [ x*(-4 * log5) + (3 * log5 + 3) ] / (log4 - 2)
Now, you have an expression for x in simplified form:
x = [ x*(-4 * log5) + (3 * log5 + 3) ] / (log4 - 2)
This is the exact value of x for which the given equation holds.