Simplify the expressions to fill in the gaps.
", "preamble": {"js": "", "css": ""}, "functions": {}, "extensions": [], "ungrouped_variables": ["x1", "y1"], "metadata": {"description": "Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.
", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "parts": [{"type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "steps": [{"type": "information", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "marks": 0, "prompt": "When adding and subtracting logarithms we can simplify the expressions using some logarithm laws. These laws are
\n\\[\\begin{align}
\\log_a(x)+\\log_a(y)&=\\log_a(xy)\\text{,}\\\\
\\log_a(x)-\\log_a(y)&=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}
\\end{align}\\]
$\\log_a(\\var{x1[1]})+ \\log_a(\\var{x1[0]})=\\log_a($ [[0]]$)$
", "gaps": [{"correctAnswerStyle": "plain", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "minValue": "x1[1]*x1[0]", "marks": "2", "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "x1[1]*x1[0]"}]}, {"type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "steps": [{"type": "information", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "marks": 0, "prompt": "When adding and subtracting logarithms we can simplify the expressions using some logarithm laws. These laws are
\n\\[\\begin{align}
\\log_a(x)+\\log_a(y)&=\\log_a(xy)\\text{,}\\\\
\\log_a(x)-\\log_a(y)&=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}
\\end{align}\\]
$\\log_a(\\var{(x1[4])*y1})-\\log_a(\\var{x1[4]})=\\log_a($ [[0]]$)$
", "gaps": [{"correctAnswerStyle": "plain", "mustBeReducedPC": 0, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "minValue": "y1", "marks": "2", "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "scripts": {}, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "y1"}]}], "tags": ["addition and subtraction of logarithms", "Laws of logarithms", "laws of logarithms", "logarithms", "Logarithms", "logs", "Logs", "taxonomy"], "advice": "We need to use the rule
\n\\[\\log_a(x)+\\log_a(y)=\\log_a(xy)\\text{.}\\]
\nSubstituting in our values for $x$ and $y$ gives
\n\\[\\begin{align}
\\log_a(\\var{x1[1]})+\\log_a(\\var{x1[0]})&=\\log_a(\\var{x1[1]}\\times \\var{x1[0]})\\\\
&=\\log_a(\\var{x1[1]*x1[0]})\\text{.}
\\end{align}\\]
\n
We need to use the rule
\n\\[\\log_a(x)-\\log_a(y)=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}\\]
\nSubstituting in our values for $x$ and $y$ gives
\n\\[\\begin{align}
\\log_a(\\var{x1[4]*y1})-\\log_a(\\var{x1[4]})&=\\log_a(\\var{x1[4]*y1}\\div \\var{x1[4]})\\\\
&=\\log_a(\\var{y1})\\text{.}
\\end{align}\\]