Q1:solve log(5)x-log(25)(x+10) = 0.5
Q2:If 2log(a)x=1+log(a) (7x-10a) ,find x in terms of a.
Q3:Find x for which 27x3^lgx = 9^1+lg(x-20)
Q4:Find x in terms of a and c ,given that log(√ a)(1/x)+log(a)x +log(a^2) x +log (a^4)x=c
since 25=5^2, log(25) = 1/2 log(5) and we have
log(5)x - 1/2 log(5)(x+1) = 0.5
or,
2log(5)x - log(5)(x+10) = 1
log(5)(x^2 / (x+10)) = 1
x^2/(x+10) = 5
x^2 = 5x+50
x^2-5x-50 = 0
(x-10)(x+5) = 0
x=10, since log(-5) is not defined.
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2log(a)x=1+log(a) (7x-10a)
since log(a)a = 1,
log(a) x^2 = log(a) (a(7x-10a))
x^2 = a(7x-10a)
x^2 = 7ax - 10a^2
x^2 - 7ax + 10a^2 = 0
(x-5a)(x-2a) = 0
x = 2a or 5a
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27x3^lgx = 9^1+lg(x-20)
Not sure what this means, but if you meant
27(3^lgx) = 9^(1+lg(x-20))
3^3 * 3^lgx = 3^(2 + 2lg(x-20))
1+lgx = 2+2lg(x-20)
lg10 + lgx = lg100 + lg(x-20)^2
10x = 100(x-20)^2
10x = 100x^2 - 4000x + 400
100x^2 - 4010x + 400 = 0
10(10x-1)(x-40) = 0
x = 1/10 or 40
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log(√a)(1/x)+log(a)x +log(a^2)x +log(a^4)x=c
2log(a)(1/x) + log(a)x + 1/2 log(a)x + 4log(a)x = c
(1/x^2)(x)(√x)(x^4) = a^c
x^(7/2) = a^c
x = a^(2/7 c)
For the Qn3 ，how u get 3^1+lgx from 3^3 *3^ lgx
？ why not 3^ 3+lgx
And the ans for Qn4 should be x=a^（-4c）...
btw,teacher,how to do this Question? log(5)(5-4x)=log(√5)(2-x) ?
Q4c:
1/x^2*x*sqrt(x)*sqrt(sqrt(x))=a^c;
1/x^(1/4) = a^c
x=a^(-4c)
For:
log5(5-4x)=log√5(2-x)
log5(5-4x)=log5(2-x)²
5-4x=(2-x)²
x=±1
Way to watch, MathMate. I missed the 1/x^2 and read it as x^2. Simpler answer, too.
Note to self: always check your answer!
Q1: To solve the equation log(5)x - log(25)(x+10) = 0.5, we can use the logarithmic properties.
Step 1: Apply the logarithmic properties to simplify the equation.
Using the rule log(a) - log(b) = log(a/b), we can rewrite the equation as:
log(5)(x / (x+10)) = 0.5.
Step 2: Convert the logarithmic equation into its exponential form.
In exponential form, the equation becomes:
(x / (x+10)) = 5^0.5.
Step 3: Simplify the exponential equation.
Since 5^0.5 is the square root of 5, we have:
(x / (x+10)) = √5.
Step 4: Solve for x.
To get rid of the fraction, cross multiply:
x = √5 * (x+10).
Expand the right side of the equation:
x = √5 * x + √5 * 10.
Now, subtract √5 * x from both sides:
x - √5 * x = √5 * 10.
Factor out x on the left side of the equation:
(1 - √5) x = √5 * 10.
Divide both sides by (1 - √5):
x = (√5 * 10) / (1 - √5).
Therefore, x is equal to (√5 * 10) / (1 - √5).
Q2: To solve the equation 2log(a)x=1+log(a) (7x-10a) for x in terms of a, we can follow these steps:
Step 1: Apply the logarithmic properties to simplify the equation.
Use the rule log(a) + log(b) = log(ab), rewrite the equation as:
log(a)x^2 = log(a)(7x-10a) + 1.
Step 2: Convert the logarithmic equation into its exponential form.
In exponential form, the equation becomes:
x^2 = (7x-10a)a + a.
Step 3: Simplify the equation by expanding and combining like terms.
Expand the right side:
x^2 = 7ax - 10a^2 + a.
Step 4: Rearrange the equation to isolate x.
Move all terms to one side of the equation:
x^2 - 7ax + 10a^2 - a = 0.
Step 5: Solve for x.
There are different methods to solve this quadratic equation, such as factoring, using the quadratic formula, or completing the square.
For example, using the quadratic formula, we have:
x = (-(-7a) ± √((-7a)^2 - 4(1)(10a^2 - a))) / (2(1)).
Simplify the equation further to get the final expression for x in terms of a.
Q3: To find x in the equation 27x * 3^lgx = 9^(1+lg(x-20)), follow these steps:
Step 1: Simplify the equation using logarithmic properties.
Apply the property log(a)b^c = c * log(a)b:
x * log(3)x = (1+log(x-20)log(9).
Step 2: Convert the logarithmic equation into its exponential form.
In exponential form, the equation becomes:
3^(log(3)x) = (x-20)^log(9)9.
Note: We can simplify further by using the fact that log(3)x = log(x) / log(3).
Step 3: Rewrite the equation.
The equation becomes:
3^(log(x)/log(3)) = (x-20) * 2.
Step 4: Solve for x.
Apply the properties of exponents to simplify the equation.
Raise both sides of the equation to the power of log(3):
(x) = (x-20)^2.
Expand the right side of the equation:
x = (x-20)(x-20).
Distribute the terms on the right side and solve for x:
x = x^2 - 40x + 400.
Rearrange to form a quadratic equation:
x^2 - 41x + 400 = 0.
Solve the quadratic equation using factoring, the quadratic formula, or other methods to obtain values for x.
Q4: To find x in terms of a and c in the equation log(√ a)(1/x) + log(a)x + log(a^2)x + log(a^4)x = c, follow these steps:
Step 1: Combine the logarithmic terms using logarithmic properties.
Using the property log(a)b + log(a)c = log(a)(b * c),
the equation becomes:
log(√ a)(1/x) * a * a^2 * a^4 = c.
Simplify the equation:
log(√ a)(1/x) * (a^8 / x) = c.
Step 2: Apply the logarithmic properties to rewrite the equation.
Using the rule log(a)b * log(b)a = 1,
the equation becomes:
(1/x) * (a^8 / x) = 10^c.
Simplify further:
(a^8) / (x^2) = 10^c.
Step 3: Solve for x.
Multiply both sides of the equation by x^2:
(a^8) = (x^2) * 10^c.
Take the square root of both sides to isolate x:
x = √(a^8 / 10^c).
Hence, x is equal to √(a^8 / 10^c) in terms of a and c.