Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Consider the equation 10^((2z)/(3))=15. Solve the equation for z. Express the solution as a logarithm in base10. z= Approximate the value of z. Round your answer to the nearest thousandth. z~~

Consider the equation 102z3=15 10^{\frac{2 z}{3}}=15 .\newlineSolve the equation for z z . Express the solution as a logarithm in base1010.\newlinez= z= \newlineApproximate the value of z z . Round your answer to the nearest thousandth.\newlinez≈ z \approx

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Compare linear and exponential growthright chevron icon

Full solution

Q. Consider the equation 102z3=15 10^{\frac{2 z}{3}}=15 .\newlineSolve the equation for z z . Express the solution as a logarithm in base1010.\newlinez= z= \newlineApproximate the value of z z . Round your answer to the nearest thousandth.\newlinez≈ z \approx
  1. Write Equation: Write down the given equation.\newlineWe have the equation 10(2z3)=1510^{\left(\frac{2z}{3}\right)} = 15.
  2. Take Logarithm: Take the logarithm of both sides of the equation to solve for zz.\newlineWe use the property that if ab=ca^b = c, then log⁡a(c)=b\log_a(c) = b.\newlineTaking the logarithm base 1010 of both sides, we get log⁡10(10(2z3))=log⁡10(15)\log_{10}(10^{(\frac{2z}{3})}) = \log_{10}(15).
  3. Simplify Left Side: Simplify the left side of the equation using the property of logarithms.\newlineThe logarithm of a power is the exponent times the logarithm of the base, so log⁡10(102z3)\log_{10}(10^{\frac{2z}{3}}) simplifies to 2z3\frac{2z}{3}.\newlineOur equation now is 2z3=log⁡10(15)\frac{2z}{3} = \log_{10}(15).
  4. Solve for z: Solve for z.\newlineTo isolate z, we multiply both sides of the equation by 32\frac{3}{2}.\newlinez=(32)⋅log⁡10(15)z = \left(\frac{3}{2}\right) \cdot \log_{10}(15).
  5. Calculate z: Calculate the value of z using a calculator.\newlineUsing a calculator, we find that log⁡10(15)\log_{10}(15) is approximately 1.1761.176.\newlineSo, z≈(32)×1.176z \approx \left(\frac{3}{2}\right) \times 1.176.
  6. Perform Multiplication: Perform the multiplication to find the approximate value of zz.\newlinez≈(32)×1.176≈1.764z \approx \left(\frac{3}{2}\right) \times 1.176 \approx 1.764.

More problems from Compare linear and exponential growth

Question
We want to factor the following expression:\newlinex4+10x+25 x^{4}+10 x+25 \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline(x−2)4−(y+3)4 (x-2)^{4}-(y+3)^{4} \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline(x−1)2+(x−1)+4 (x-1)^{2}+(x-1)+4 \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline9x4+4y2 9 x^{4}+4 y^{2} \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline16x4−25 16 x^{4}-25 \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newlinex4+3x2y+9y2 x^{4}+3 x^{2} y+9 y^{2} \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline(x+5)4+2(x+5)2−1 (x+5)^{4}+2(x+5)^{2}-1 \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline(x−3)2−10(x−3)+25 (x-3)^{2}-10(x-3)+25 \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline(x+7)2+2y(x+7)+y2 (x+7)^{2}+2 y(x+7)+y^{2} \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

Question
We want to factor the following expression:\newline9x6+6x3y+y2 9 x^{6}+6 x^{3} y+y^{2} \newlineWhich pattern can we use to factor the expression?\newlineU U and V V are either constant integers or single-variable expressions.\newlineChoose 11 answer:\newline(A) (U+V)2 (U+V)^{2} or (U−V)2 (U-V)^{2} \newline(B) (U+V)(U−V) (U+V)(U-V) \newline(C) We can't use any of the patterns.
Get tutor helpright-arrow

Posted 10 months ago

View all (27)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant